SAN FRANCISCO—The pieces are set on the chessboard that is the Gulf of Mexico (GOM) shelf as operators turn to execution to boost reserves in a consolidated mature environment.

“Some small fields are out there, but the big ones we own, Apache owns, or Chevron owns—and none of us are selling,” said Bill Marchive, executive vice president of exploration and production for Energy XXI (Nasdaq: EXXI), presenting at the IPAA OGIS West conference in San Francisco.

“We wish Chevron would sell a big package like ExxonMobil, but they are getting ...