Senator John Thune of South Dakota, the second-ranking Senate Republican and a potential future leader, is seriously considering retiring after next year, a prospect that has set off an intensifying private campaign from other Republicans urging him to seek re-election.

Mr. Thune is only 60, but a combination of family concerns and former President Donald J. Trump’s enduring grip on the Republican Party have prompted the senator, who is in his third term, to tell associates and reporters in his home state that 2022 could be his last year in Congress…

Among those alarmed is Mr. McConnell himself, who one adviser said had “leaned in” on pushing Mr. Thune to run again.

“I certainly hope that he will run for re-election, and that’s certainly what I and others have been encouraging him to do all year long,” Mr. McConnell said in an interview.