Some members were less concerned.

“I’m praying for it every day,” Senator David Perdue, Republican of Georgia, said of the health care legislation.

In fact, a handful of wavering Republicans in the House seemed to take heart in recent days that revisions might come in the Senate.

Others have chafed at the suggestion that the upper chamber would be responsible for cleanup duty.

“They tell us they’re so smart and they’re so good at this stuff and we’re so incapable that they need to work on it — and I agree with them,” Representative Michael C. Burgess, Republican of Texas, said last week. “They need to work on it. I want them to. I’m anxious to see what they’re going to do with finishing the job that we started.”

Senator Lamar Alexander, Republican of Tennessee and the chairman of the Committee on Health, Education, Labor and Pensions, said he would “review” the House bill before proceeding.