In Washington, many of the party’s top operatives believe that there is no way even the strongest Senate candidates could overcome the tide if Mr. Trump were leading the ticket.

“Senator Portman is a great example I like to use when talking about this,” said Brian Walsh, a Senate campaign veteran. “He’s very well prepared, has tons of cash in the bank, and he got his campaign organized and up and running early. But if we nominate a bad presidential candidate like Trump, senators like Portman or Kelly Ayotte aren’t going to be able to outrun Hillary by that much. And there goes the Senate.”

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.”

Mr. Graham recounted separate phone calls with two of the party’s most sought-after donors last week, people who he insisted not be named but who give tens of millions of dollars to Republicans every election year. He said they had expressed alarm at Mr. Trump’s durability and asked what could be done.