More than 70 Republicans have signed a letter to the party's National Committee head urging him to stop helping Donald Trump's campaign.
They said Mr Trump's "divisiveness" and "incompetence" risked drowning the party in November's election.
The letter said that the party should instead focus on protecting vulnerable candidates in elections to the Senate and the House of Representatives.
Former members of Congress are among the signatories of the letter.
"We believe that Donald Trump's divisiveness, recklessness, incompetence, and record-breaking unpopularity risk turning this election into a Democratic landslide," said a draft of the letter obtained by Politico.
Options for anti-Trump Republicans
- Hope he quits: Mr Trump leaving on his own accord would be the cleanest way to replace him on the ballot, but he is unlikely to do this.
- Find another candidate to rally round: Without Mr Trump leaving the race, a replacement is impossible under party rules. Some are considering endorsing Libertarian candidate Gary Johnson, but this would split the Republican vote.
- Force him out, using an obscure rule: Some are thinking of declaring Mr Trump "not of sound mind", under a Republican National Committee manoeuvre that has not been used before. But this means calling a man chosen by more than 10 million party members insane - a bold move.
- Sit and wait: Some disaffected Republicans think their best option is to just denounce Mr Trump and hope for better luck next time.
- Put your head down and hope he wins: Others hope that he moderates his rhetoric, so that once he is elected mainstream Republicans can right the ship. However Mr Trump may have done lasting damage to the brand to stop the party's slide.
"Only the immediate shift of all available RNC (Republican National Convention) resources to vulnerable Senate and House races will prevent the GOP (Republican Party) from drowning with a Trump-emblazoned anchor around its neck."
The letter added: "This should not be a difficult decision, as Donald Trump's chances of being elected president are evaporating by the day."
Reacting to the move, Mr Trump said he was not concerned that the party could cut him off.
"All I have to do is stop funding the Republican Party," the billionaire said.
Time Magazine on Thursday reported that RNC Chairman Reince Priebus had threatened to withdraw funding from the Trump campaign, and instead direct it to Congressional campaigns.
Mr Trump denies that this conversation ever took place.
The Republican presidential nominee has endured 10 days of negative headlines after a string of controversial comments.
In recent weeks, several leading Republicans have deserted Mr Trump over his outspoken attacks.
Polls suggest support for the embattled candidate has been falling in key battleground states in recent weeks.