Still, Republicans said they did not detect a significant shift that would pose a serious threat to the president in the Senate. It would require 20 Republicans to side with Democrats in convicting Mr. Trump, and few observers believe that will happen.

Mr. McConnell, his allies said, regards the impeachment fight in much the same way as he did the struggle last year to confirm Judge Brett M. Kavanaugh to the Supreme Court, in which he was primarily concerned with protecting his Senate majority by insulating vulnerable incumbents. Then, as now, they said, Mr. McConnell is focused on keeping Republicans as united as possible, while allowing those with reservations about Mr. Trump’s conduct and their own political considerations to justify their decision to their constituents.

“I think he will play it straight,” said Senator John Cornyn, Republican of Texas and a close McConnell ally, who noted his party’s narrow voting margin. “I don’t think he has any alternative. When you are operating with 53 you have thin margins and you can’t jam anybody or you end up with undesirable consequences.”