Mr. McConnell has instructed fellow Republicans privately that they must figure out individually the impeachment message that works best for them politically. But he is also keenly focused on tailoring the process to insulate his most vulnerable members from a constituent backlash. The resolution introduced on Thursday was in part an effort to allow Republicans to unite publicly behind a measure critical of the inquiry, a way to show the party base that they were behind Mr. Trump even as they refrained from defending his actions.

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Ms. Ernst, Ms. McSally, and Senator Thom Tillis of North Carolina were among those who have so far publicly supported the measure.

Senator Todd Young of Indiana, the chairman of Senate Republicans’ campaign arm, said in a brief interview that the focus on impeachment gives senators “an opportunity” to show voters that while they are being “attentive and conscientious to current events, they remain intently focused” on the issues their constituents sent them to work on.

Campaign consultants have stressed to senators the importance of maintaining their own credibility, according to two senior Republican officials, especially given that new revelations may still emerge. They have instructed senators not to respond to every turn of the screw, one reason that most of them have dodged questions about Mr. Trump’s conduct or resorted to complaints about the process.

Mr. McConnell, for his part, offered a stilted if telling response on Tuesday, when a reporter asked him during a diplomat’s testimony this week if he was “willing to defend the president in this matter.”

“I’m willing to talk about the process in the House,” Mr. McConnell said. “I just did. I think it’s grossly unfair and I think the president has a legitimate complaint about the process.”

Some endangered Republicans have also carefully eyed blueprints tested by senators considered to be among the savviest in the conference. Senators Rob Portman of Ohio and Pat Toomey of Pennsylvania have offered one alternative: staking out the position in interviews with local publications that while Mr. Trump’s call with President Volodymyr Zelensky of Ukraine may have been inappropriate, it is not impeachable.