If Mr. Trump does not change course, his silence could effectively doom any hopes Mr. Corker has for seeking a third term. The president is highly popular among Tennessee Republicans and may be the only person who could reverse Mr. Corker’s standing with primary voters, who have soured on him since he portrayed Mr. Trump as a juvenile in need of day care, whose instability may push the country into World War III.

And without Mr. Trump’s direct intervention, Ms. Blackburn is highly unlikely to bow to Mr. Corker.

“Marsha Blackburn is not getting out of this race regardless of who gets in,” said Ward Baker, the Nashville-area lawmaker’s chief strategist.

Further, Senator Mitch McConnell of Kentucky, the majority leader, has rebuffed Mr. Corker by telling him that he must secure the president’s support to re-enter the race, according to Republicans familiar with the conversation, a rare act of political deference that suggests he is uneasy about driving Ms. Blackburn out of the primary race.

But Mr. Corker and some of his Senate allies are aggressively working to win over the White House, embarking on what one West Wing official described as a sudden charm offensive. The senator has avoided any criticism of Mr. Trump in recent weeks and on Monday met with Ivanka Trump, the president’s daughter.

Mr. Trump’s political advisers, getting wind of the meeting, scrambled to brief Ms. Trump and her own staff about Mr. Corker’s renewed interest in running again and his desire for the president’s support, according to a Republican official. An aide to Mr. Corker said Ms. Trump requested the meeting.