“It’s, uh, well, look at the alternatives,” Senator Jeff Flake of Arizona said when asked if his party could rally around Mr. Cruz at some juncture. Mr. Cruz has been gaining on Mr. Trump over the last week and lags him by fewer than 100 delegates, while a new poll shows Mr. Trump’s popularity may be flagging after a wave of new attacks ads.

“I am crestfallen that my candidate, Jeb Bush, is out of the running,” said Senator Susan Collins of Maine, who added that she would endorse no one.

Perhaps she was just being self-interested. Ms. Collins joked that she and Senator Joe Donnelly, Democrat of Indiana, having recently been named the two most bipartisan senators by a policy think tank, were planning to run on their own ticket. “We were worried about having an all-Catholic ticket,” Ms. Collins said.

Mr. Cruz is one of the least popular senators on Capitol Hill, in large part because he masterminded the 2013 government shutdown in a failed attempt to roll back President Obama’s health care law, something Senator Richard M. Burr, Republican of North Carolina, said at the time was “the dumbest idea I have ever heard.”