how do you solve a problem like The Donald?

For months, much of the Republican Party’s establishment has been uneasy about the rise of Donald J. Trump, concerned that he was overwhelming the presidential primary contest and encouraging other candidates to mimic his incendiary speech. Now, though, irritation is giving way to panic as it becomes increasingly plausible that Mr. Trump could be the party’s standard-bearer and imperil the careers of other Republicans.

Many leading Republican officials, strategists and donors now say they fear that Mr. Trump’s nomination would lead to an electoral wipeout, a sweeping defeat that could undo some of the gains Republicans have made in recent congressional, state and local elections. But in a party that lacks a true leader or anything in the way of consensus — and with the combative Mr. Trump certain to scorch anyone who takes him on — a fierce dispute has arisen about what can be done to stop his candidacy and whether anyone should even try.

Some of the highest-ranking Republicans in Congress and some of the party’s wealthiest and most generous donors have balked at trying to take down Mr. Trump because they fear a public feud with the insult-spewing media figure. Others warn that doing so might backfire at a time of soaring anger toward political insiders.

That has led to a standoff of sorts: Almost everyone in the party’s upper echelons agrees something must be done, and almost no one is willing to do it.

With his knack for offending the very constituencies Republicans have struggled with in recent elections, women and minorities, Mr. Trump could be a millstone on his party if he won the nomination. He is viewed unfavorably by 64 percent of women and 74 percent of nonwhite voters, according to a November ABC News/Washington Post poll. Such unpopularity could not only doom his candidacy in November but also threaten the party’s tenuous majority in the Senate, hand House seats to the Democrats and imperil Republicans in a handful of governor’s races.

Asked about concerns over Mr. Trump’s potential influence on other contests, his spokeswoman, Hope Hicks, said, “I think the facts indicate the exact opposite is true,” and emailed a link to a consumer marketing firm’s assertion that Mr. Trump would ensure the highest general election turnout from Republicans, Democrats and independents alike.

Yet the clamor for a “Stop Trump” effort has become pervasive at the Senate’s highest levels, where members up for re-election are realizing that they can no longer dismiss as strictly theoretical the possibility of his capturing the nomination. Mr. Trump’s persistent ranking at or near the top of the polls is prompting urgent calls for an advertising assault to try to sink his campaign.

“It would be an utter, complete and total disaster,” Senator Lindsey Graham of South Carolina, himself a presidential candidate who has tangled with Mr. Trump, said of his rival’s effect on lower-tier Republican candidates. “If you’re a xenophobic, race-baiting, religious bigot, you’re going to have a hard time being president of the United States, and you’re going to do irreparable damage to the party.”

Slowly, some members of the party’s establishment are reckoning with the idea of a Trump ticket. The National Republican Senatorial Committee has cautioned its incumbents in blunt terms not to let themselves be linked to him.

But beyond sheer intimidation, some members of Congress worry that if the party’s establishment went after Mr. Trump, it would only fuel his anti-Washington appeal.

“I think it would play into his hands and only validate him,” said Senator Lamar Alexander, Republican of Tennessee. “A ‘Stop Trump’ effort wouldn’t work, and it might help him.”