TRANSOCEAN OFFSHORE INC
4, 1998-06-11
DRILLING OIL & GAS WELLS
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UNITED STATES SECURITIES AND EXCHANGE COMMISSION
WASHINGTON, D.C. 20549
FORM 4
STATEMENT OF CHANGES IN BENEFICIAL OWNERSHIP
( ) Check this box if no longer subject to Section 16.
Form 4 or Form 5 obligations may continue. See Instructions 1(b).
1. Name and Address of Reporting Person
   Cole, Jon C.
   4 Greenway Plaza
   Houston, Texas  77046
   USA
2. Issuer Name and Ticker or Trading Symbol
   Transocean Offshore Inc.
   RIG
3. IRS or Social Security Number of Reporting Person (Voluntary)
   
4. Statement for Month/Year
   June 1, 1998
5. If Amendment, Date of Original (Month/Year)
   
6. Relationship of Reporting Person(s) to Issuer (Check all applicable)
   ( ) Director ( ) 10% Owner (X) Officer (give title below) ( ) Other
   (specify below)
   Senior Vice President
7. Individual or Joint/Group Filing (Check Applicable Line)
   (X) Form filed by One Reporting Person
   ( ) Form filed by More than One Reporting Person
<TABLE>
<CAPTION>
___________________________________________________________________________________________________________________________________
 Table I -- Non-Derivative Securities Acquired, Disposed of, or Beneficially Owned                                                 |
___________________________________________________________________________________________________________________________________|
1. Title of Security       |2.    |3.    |4.Securities Acquired (A)         |5.Amount of        |6.Dir |7.Nature of Indirect       |
                           | Transaction |  or Disposed of (D)              |  Securities       |ect   |  Beneficial Ownership     |
                           |      |      |                                  |  Beneficially     |(D)or |                           |
                           |      |    | |                  | A/|           |  Owned at         |Indir |                           |
                           | Date |Code|V|    Amount        | D |    Price  |  End of Month     |ect(I)|                           |
___________________________________________________________________________________________________________________________________|
<S>                        <C>    <C>  <C><C>               <C> <C>         <C>                 <C>    <C>
Common Stock               |5/9/98|M   | |70,000            |A  |$9.25      |                   |      |                           |
- -----------------------------------------------------------------------------------------------------------------------------------|
Common Stock               |5/698 |M   | |30,000            |A  |$8.375     |                   |      |                           |
- -----------------------------------------------------------------------------------------------------------------------------------|
Common Stock               |5/6/98|S   | |100,000           |D  |$57.4034   |54,828 (1)         |D     |                           |
- -----------------------------------------------------------------------------------------------------------------------------------|
Common Stock               |      |    | |                  |   |           |6,144              |I     |(2)                        |
- -----------------------------------------------------------------------------------------------------------------------------------|
___________________________________________________________________________________________________________________________________|
<CAPTION>
___________________________________________________________________________________________________________________________________
 Table II -- Derivative Securitites Acquired, Disposed of, or Beneficially Owned                                                   |
___________________________________________________________________________________________________________________________________|
1.Title of Derivative |2.Con-  |3.   |4.    |5.Number of De |6.Date Exer|7.Title and Amount  |8.Price|9.Number    |10.|11.Nature of|
  Security            |version |Transaction | rivative Secu |cisable and|  of Underlying     |of Deri|of Deriva   |Dir|Indirect    |
                      |or Exer |     |      | rities Acqui  |Expiration |  Securities        |vative |tive        |ect|Beneficial  |
                      |cise    |     |      | red(A) or Dis |Date(Month/|                    |Secu   |Securities  |(D)|Ownership   |
                      |Price of|     |      | posed of(D)   |Day/Year)  |                    |rity   |Benefi      |or |            |
                      |Deriva- |     |      |               |Date |Expir|                    |       |ficially    |Ind|            |
                      |tive    |     |      |           | A/|Exer-|ation|   Title and Number |       |Owned at    |ire|            |
                      |Secu-   |     |    | |           | D |cisa-|Date |   of Shares        |       |End of      |ct |            |
                      |rity    |Date |Code|V|  Amount   |   |ble  |     |                    |       |Month       |(I)|            |
___________________________________________________________________________________________________________________________________|
<S>                   <C>      <C>   <C>  <C><C>        <C> <C>   <C>   <C>          <C>     <C>     <C>          <C> <C>
Executive Stock Option|$9.25   |5/6/9|M   | |70,000     |D  |(3)  |8/31/|Common Stock|70,000 |       |0           |D  |            |
s                     |        |8    |    | |           |   |     |04   |            |       |       |            |   |            |
- -----------------------------------------------------------------------------------------------------------------------------------|
Executive Stock Option|$8.375  |5/6/9|M   | |30,000     |D  |(4)  |1/28/|Common Stock|30,000 |       |50,000      |D  |            |
s                     |        |8    |    | |           |   |     |04   |            |       |       |            |   |            |
- -----------------------------------------------------------------------------------------------------------------------------------|
                      |        |     |    | |           |   |     |     |            |       |       |            |   |            |
___________________________________________________________________________________________________________________________________|
</TABLE>
Explanation of Responses:
(1) Includes shares  of unvested restricted stock granted under the Transocean 
Offshore Inc. Long-Term Incentive Plan ("LTIP").  
(2) Shares held under Transocean Offshore Savings Plan at 12/31/97.  The number 
of shares is approximate because Plan accounting is on a unit 
basis.
(3) Grant of 100,000 options under the LTIP exercisable as follows; 100,000 on 
6/21/95.
(4) Grant of 80,000 options under the LTIP exercisable as follows; 26,666 on 
1/28/95, 26,668 on 1/28/96 and 26,666 on 1/28/97.
                                                  
SIGNATURE OF REPORTING PERSON
/s/ Jon C. Cole by Nicolas J. Evanoff
DATE
June 11,  1998



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