WASHINGTON—Exxon Mobil Corp. has applied to the Treasury Department for a waiver from U.S. sanctions on Russia in a bid to resume its joint venture with state oil giant PAO Rosneft, according to people familiar with the matter.

Exxon has been seeking U.S. permission to drill with Rosneft in several areas banned by sanctions and renewed a push for approval in March, shortly after its most recent chief executive, Rex Tillerson, became secretary of state on Feb. 1, according to one of these people. The company originally applied...