WASHINGTON—President Donald Trump, whose family and political aides have faced scrutiny over their ties to Russia, rejected a bid by Exxon Mobil Corp. to sidestep U.S. sanctions against Moscow and resume an oil venture with a politically powerful Russian energy firm.

The announcement Friday comes as the White House pushes to firm up the president’s foreign-policy and domestic agenda as he nears his 100th day in office next week.

Mr....