“When you do that, it hurts the credibility of the presidency,” Mr. Christie said on CNBC.

A Monmouth University poll of New Hampshire voters released last week emphasized the problem for them: With Mr. Trump in the race, Mr. Christie drew 4 percent and Gov. Scott Walker of Wisconsin 7 percent. But the same poll showed that, if Mr. Trump were to drop out, Mr. Christie would pick up 15 percent of Trump voters, and Mr. Walker 14 percent.

In Iowa, a crucial state for Mr. Walker, Mr. Trump is now statistically tied with him, according to an NBC Marist poll of potential Republican caucusgoers.

Mr. Bush’s enthusiasts, meanwhile, are all but giddy, even though it would be much better if he were simply dominating the field.

“You got to give Trump credit,” Senator Orrin G. Hatch, Republican of Utah, said with a grin. “He’s really created a firestorm here. Because he is such a controversial character, and an interesting one at that, it’s taken away opportunities for an awful lot of others to break through.” At the same time, Mr. Hatch said, Mr. Bush “comes across as the one guy you can rely on.”

Mr. Bush has been openly critical of Mr. Trump and “the vitriol that he’s spewing out these days,” while being increasingly emphatic about the need for Republicans to embrace a more optimistic and inclusive message. It is a theme aimed not just at the general electorate, but also at Republican primary voters eager to back a candidate who can generate broad appeal and win the White House.