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                                                                   EXHIBIT 10c
 
                             ROWAN COMPANIES, INC.
 
                   1998 CONVERTIBLE DEBENTURE INCENTIVE PLAN
 
     1. Purpose. The Rowan Companies, Inc. 1998 Convertible Debenture Incentive
Plan (the "Plan") is intended to promote the interests of Rowan Companies, Inc.
(the "Company") and its stockholders by allowing officers and other key
personnel of the Company and its subsidiaries the opportunity to invest in
corporate debt in the form of the Company's floating interest rate subordinated
debentures (the "Debentures") which are convertible into shares of Preferred
stock, $1 par value, of the Company (the "Preferred Stock"), which shares of
Preferred Stock are convertible into shares of common stock of the Company (the
"Common Stock"), thereby giving key personnel added incentive to work toward the
continued growth and success of the Company. The Company's Board of Directors
also contemplates that the Plan will enable the Company and its subsidiaries to
compete more effectively for the services of management personnel needed for the
continued growth and success of the Company.
 
     2. Issuance of the Debentures. The Company shall have authority to issue
Debentures in such amounts and to such of the key employees of the Company and
its subsidiaries (as defined by Section 425 of the Internal Revenue Code of
1954, as amended) as the Committee (as defined in Section 9) shall from time to
time determine. Such employees purchasing Debentures are designated herein as
"Purchasers".
 
     3. General Terms and Conditions of the Debentures.
 
     Section 3.1. General. The Committee shall from time to time determine with
respect to each series of Debentures to be issued the interest rate thereof, the
conversion price applicable thereto (including the conversion ratio of the
Preferred Stock), and such other terms and conditions of the Debentures, all to
the extent not inconsistent with the provisions of this Plan.
 
     Section 3.2. Form and Term of Debentures. Debentures will be issued in
series, the terms and conditions of which may differ among series and shall be
in such form and in such denominations as the Committee may approve. Each series
will be due not earlier than five years, or later than ten years, from the date
of issuance, or on such earlier date as the Company redeems any Debentures,
which date is referred to herein as the "Due Date".
 
     Section 3.3. Conversion of the Debentures. Subject to the provisions of
this Section 3.3, the Debentures will be convertible at the conversion price in
effect at the time of conversion into fully paid and non-assessable shares of
Preferred Stock, which will immediately be convertible into fully paid and
non-assessable shares of Common Stock of the Company, at any time in portions
and after time periods determined by the Committee, which in no event will be
less than one year after the date of issuance until the close of business on the
Due Date. Each series of Debentures shall be convertible into a separate series
of Preferred Stock. The conversion privilege with respect to any Debenture may
be exercised only by a Purchaser thereof, by the estate of a deceased Purchaser
or a beneficiary under such estate, or by a transferee of the Debenture as
provided for therein.
 
     Upon termination of a Purchaser's employment except as described in (a),
(b), (c) or (d) below, a Purchaser's conversion privilege shall terminate. Upon
termination of employment as described in any of (a), (b), (c) or (d) below, the
Purchaser may thereafter exercise his conversion privilege at any time prior to
the earlier of the Due Date or:
 
          (a) If Employee's employment with the Company terminates by reason of
     Retirement (as defined in Appendix A hereto), the date which is five years
     following the date of such Retirement, but only as to the portion or
     portions of the Debenture as provided for therein that Purchaser was
     entitled to convert as of the date of such Retirement, plus such additional
     portion or portions, if any, that the Committee, in its sole discretion,
     determines to be convertible as of the date of such Retirement.
 
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          (b) If Employee dies within the five-year period following the date of
     Employee's termination of employment by reason of Retirement (in which case
     conversion may be effected as applicable by Employee's estate, or the
     person who acquires this Debenture by bequest or inheritance or otherwise
     by reason of the death of Employee), the date which is two years following
     the date of Employee's death, but only as to the portion or portions of the
     Debenture as provided for therein that Purchaser was entitled to convert as
     of the date Employee's employment terminated by reason of Retirement.
 
          (c) If Employee's employment with the Company terminates by reason of
     Disability (as defined in Appendix A hereto), Employee may convert this
     Debenture in full at any time during the period of five years following the
     date of such termination.
 
          (d) If Employee dies while in the employ of the Company or within the
     five-year period following the date of Employee's termination of employment
     by reason of Disability, Employee's estate, or the person who acquires this
     Debenture by bequest or inheritance or by reason of the death of Employee,
     may convert this Debenture in full at any time during the period of two
     years following the date of Employee's death.
 
     In the case of (a), (b), (c) and (d), any and all questions as to whether
and when there has been a termination of a Purchaser's employment, and the cause
of such termination, shall be determined by the Committee using whatever
evidence as it may require, such determination or determinations being final and
controlling on all interested parties.
 
     The conversion privilege with respect to any Debenture (i) will terminate
if the Purchaser, without the Company's consent, sells, assigns, transfers,
pledges, hypothecates or otherwise disposes of a Debenture except as permitted
by Section 3.4 and (ii) will not be exercisable during such time as the
Debenture is pledged to secure loans as permitted by Section 3.4.
 
     In no event may any Purchaser or the estate of a deceased Purchaser or a
beneficiary under such estate exercise the conversion privilege associated with
a Debenture prior to one year from the date of issuance of such Debenture or
after the Due Date.
 
     Notwithstanding the foregoing or any provision in the Plan to the contrary,
effective with the occurrence of a Corporate Change, as defined in the Company's
Restated 1988 Nonqualified Stock Option Plan, each Debenture that has been
issued and outstanding for more than one year as of the date of the Corporate
Change shall automatically be fully convertible on and after the date of such
Corporate Change.
 
     Section 3.4. Transfer and Pledge of Debentures. A Purchaser may not sell,
assign, transfer, pledge, hypothecate or otherwise dispose of a Debenture except
by (i) will or the laws of descent and distribution or (ii) a pledge ("Permitted
Pledge") of Debentures to a lender (which may be the Company if a loan is made
pursuant to Section 8 hereof) as security for loans to provide all or part of
the financing to purchase the Debentures. If such loan shall be made by other
than the Company, the Purchaser shall give advance written notice to the Company
prior to making any Permitted Pledge and the Purchaser and such Lender shall
give notice of discharge of any Debenture from a Permitted Pledge, which notice
shall be conclusive evidence that the conversion privilege with respect to such
Debenture will again be exercisable subject to the provisions of Section 3.3.
 
     Section 3.5. Redemption of Debentures. The Company may, upon at least
thirty days prior written notice to the affected Debenture holder or holders,
redeem on any interest payment date, any or all of the Debentures issued under
this Plan. The Company shall redeem on the next interest payment date after
termination of the conversion privilege with respect thereto any Debenture with
respect to which the conversion privilege has terminated pursuant to clauses
(a), (b), (c) or (d) of Section 3.3. The holder of any Debenture redeemed
pursuant to this Section 3.5 shall be entitled to receive only the face amount
of the Debenture plus accrued interest thereof to the Due Date.
 
     4. Authorized Amount of Debentures. The Company may issue up to
$30,000,000.00 in aggregate principal amount of all Debentures.
 

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     5. Effective Date. The Plan shall become effective upon approval thereof by
the vote of the holders of a majority of the shares of Common Stock of the
Company voting at the 1998 Annual Meeting of Stockholders, and shall expire when
all of the Company's obligations with respect to all of the outstanding
Debentures have been discharged; provided, however, that no Debenture shall be
issued after April 24, 2008.
 
     6. Offers and Sales Price of Debentures. The Debentures shall be sold by
the Company to Purchasers at a price equal to the higher of (a) face value plus
any accrued interest to the date of sale or (b) the fair market value of the
Debentures as of the date the Purchaser elects to purchase the Debentures, as
determined by an independent investment banking firm. If the Internal Revenue
Service determines that the value of a Debenture at the time of sale exceeded
its sale price and if (a) the Company receives a federal income tax benefit as a
result of such determination and (b) the Purchaser has contested such
determination in a manner which the Company determines to be appropriate under
the circumstances, then the Company will pay to the Purchaser or his estate or a
beneficiary under his estate the lesser of (x) the federal income tax benefit
derived by the Company as a result of the sale of the Debenture to the Purchaser
or (y) the amount estimated by the Company (based on the highest marginal
federal income tax rate applicable with respect to compensation income for the
year in which the sale occurred and the amount determined by the Internal
Revenue Service to be taxable income to the Purchaser as a result of his
purchase of the Debenture) to be Purchaser's federal income tax liability
resulting from his purchase of the Debenture.
 
     The Debentures may be offered only on the dates of regularly-scheduled
meetings of the Company's Board of Directors (any such date is referred to
herein as an "Offering Date"). An employee may elect to purchase all or none of
the Debentures offered to him on an Offering Date by giving written notice to
the Company of his election within 10 business days of such Offering Date.
Payment for such Debentures shall be in cash or in Common Stock (valued at the
mean of the high and low sales price of Common Stock prior to the date of such
payment, as shown on the Composite Tape for securities listed on the New York
Stock Exchange) and shall be made within 20 business days of such Offering Date.
 
     7. Conversion Price. The price (the "Conversion Price") at which shares of
Preferred Stock shall be delivered upon conversion of a series of Debentures
shall be set at a price at least equal to the mean of the high and low sales
price of the Company's Common Stock on the date of sale of such series of
Debentures, as shown on the Composite Tape for securities listed on the New York
Stock Exchange. The number of shares of Common Stock which shall be delivered
upon conversion of any shares of a series of Preferred Stock (the "Conversion
Ratio") shall not exceed the face value of the related Debentures which were
converted into such Preferred Stock divided by the mean of the high and low
sales price of the Company's Common Stock on the date of sale of such series of
Debentures, as shown on the Composite Tape for securities listed on the New York
Stock Exchange. Upon any change in the capital stock of the Company, through
merger, consolidation, reorganization, recapitalization, stock dividend, stock
split, spin-off, split up, dividend in kind or other change in the corporate
structure or distribution to stockholders, appropriate adjustments to the
Conversion Price and Conversion Ratio and the kind of shares delivered upon
conversion of the Debentures and Preferred Stock may be made by the Committee
(or, if the Company is not the surviving corporation in any such transaction,
the board of directors of the surviving corporation) with respect to both
outstanding and unissued Debentures and Preferred Stock. If the Internal Revenue
Service determines that the conversion of Debentures into Preferred Stock or
that the subsequent conversion of Preferred Stock into Common Stock is a taxable
transaction and if (a) the Company receives a federal income tax benefit as a
result of such determination and (b) the Purchaser has contested such
determination in a manner which the Company deems to be appropriate under the
circumstances, then the Company will pay the Purchaser or his estate or a
beneficiary under his estate the lesser of (x) the federal income tax benefit
derived by the Company with respect to such conversion or (y) the amount
estimated by the Company (based on the highest marginal federal income tax rate
applicable with respect to compensation income for the year in which the
conversion occurred and the amount determined by the Internal Revenue Service to
be taxable income to the Purchaser as a result of such conversion) to be
Purchaser's federal income tax liability resulting from such conversion.
 
     8. Company Loans. The Company may, from time to time, make loans ("Company
Loans") to Purchasers for the purpose of providing all or part of the financing
necessary to purchase any Debenture; provided, however, that the maximum amount
of the Company Loan shall not exceed the purchase price of


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the Debentures. Subject to the foregoing, Company Loans may be made to such
Purchasers in such amounts bearing interest at such rates (not less than the
higher of the interest rate on the Debenture or a floating rate determined under
Sections 483 and 1274(d) of the Internal Revenue Code of 1986, as amended),
shall be secured by a pledge of and lien on the Debenture (which may be inferior
to the pledge and lien securing a bank loan) and on such other terms and
conditions as the Committee may from time to time approve.
 
     9. Administration. The Plan shall be administered by a committee of the
Board of Directors (the "Committee"), which shall consist of three or more
persons. No Debentures may be sold to any member of the Committee during the
term of his membership on the Committee. No person shall be eligible to serve on
the Committee unless he is a "disinterested person" within the meaning of
Paragraph (d)(3) of Rule 16b-3, under the Securities Exchange Act of 1934 or any
successor thereto as then in effect ("Rule 16b-3"). The members of the Committee
shall be appointed by the Board of Directors, and any vacancy on the Committee
shall be filled by the Board of Directors.
 
     Subject to the foregoing paragraphs, the Committee shall interpret the Plan
and the Debentures sold under the Plan, shall make such rules as it deems
necessary for the proper administration of the Plan, shall make all other
determinations necessary or advisable for the administration of the Plan and
shall correct any defect or supply any omission or reconcile any inconsistency
in the Plan or in any Debenture in the manner and to the extent the Committee
deems desirable to administer the Plan or the Debentures. The Committee's
determination of any matter within its authority shall be conclusive and binding
upon the Company and all other persons.
 
     10. Amendment and Discontinuance. Subject to the provisions of this Section
10, the Committee may amend, suspend or terminate the Plan. No amendment,
suspension or termination of the Plan may:
 
          (a) Without the consent of the holder of a Debenture, terminate his
     Debenture or adversely affect his rights under the Debenture in any
     material respect;
 
          (b) Without the consent of a majority of the shares of voting stock of
     the Company voting at any meeting of Stockholders (i) increase the amount
     of Debentures available under the Plan, (ii) change materially the persons
     eligible to purchase Debentures under the Plan, (iii) increase materially
     the benefits under the Plan, or (iv) extend the termination date of the
     Plan; or
 
          (c) Cause the plan to fail to meet the requirements of Rule 16b-3.
 
     11. Other Provisions.
 
     (a) The Purchaser of a Debenture shall not be entitled to any rights as a
stockholder of the Company until such Purchaser has exercised the conversion
privilege contained in the Debenture.
 
     (b) No Debenture shall be construed as limiting any right which the Company
or any subsidiary of the Company may have to terminate at any time, with or
without cause, the employment of a Purchaser to whom a Debenture has been sold.
 
     (c) Notwithstanding any provision of the Plan or the terms of any Debenture
sold pursuant to the Plan, (i) the Company shall not be required to issue any
Debentures hereunder if such issuance would, in the judgment of the Committee,
constitute a violation of any state or Federal law, or of the rules or
regulations of any governmental regulatory body, and (ii) any amount of interest
paid or payable on a Debenture which exceeds the amount legally payable to a
Purchaser under the applicable usury laws will be paid by the Company as
compensation to the Purchaser.
 

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                                   APPENDIX A
                                       TO
                   1998 CONVERTIBLE DEBENTURE INCENTIVE PLAN
 
     Retirement. For purposes of the Plan, a Purchaser shall be deemed to have
terminated employment by reason of "Retirement" if such Purchaser voluntarily
terminates employment on or after having attained the age of 60 while employed
by the Company or an affiliate of the Company.
 
     Disability. For purposes of the Plan, a Purchaser shall be deemed to have
terminated his employment with the Company or an affiliate of the Company by
reason of "Disability" if at the time of such termination of employment he has a
mental or physical condition which totally and presumably permanently prevents
him from engaging in any substantial gainful employment with the Company which
(i) did not arise while engaged in or as a result of being engaged in an illegal
act or enterprise, (ii) did not result from chronic alcoholism, addiction to
narcotics or the use of illegal or unauthorized drugs in any manner, (iii) did
not result from service in the Armed Forces of the United States which entitled
the Purchaser to a veteran's disability pension, and (iv) did not arise while
employed by an employer other than the Company or a subsidiary or affiliated
corporation of the Company. The existence of such Disability must be certified
by two duly licensed and practicing physicians selected, respectively, at the
direction of the Committee and at the direction of the Purchaser or his
representative. If they fail to agree, a third physician shall be selected at
the direction of the Committee, and the determination of any two of such three
physicians shall be final and controlling on all interested parties. The
determination of any such physicians shall be evidenced by appropriate written
certifications delivered to the Committee. Notwithstanding the foregoing, the
Committee may, in its discretion, waive the requirement for certification of
Disability by licensed physicians, and, in lieu of such certification, rely on
such other appropriate medical evidence of Disability as is deemed satisfactory
by the Committee.