The real estate magnate has insisted he would never give up the White House race and wrote on Twitter on Sunday that Republicans who are attacking him are "self-righteous hypocrites."
With only a month to go before the Nov. 8 election against Democrat Hillary Clinton, it would be a massive stretch to substitute Trump for his running mate Mike Pence or anyone else, Republican strategists and U.S. election experts said.
It is “virtually impossible to replace somebody in August, let alone October,” Republican strategist Karl Rove said on "Fox News Sunday."
After a video of Trump making vulgar comments about women surfaced Friday, Republicans such as U.S. Senator Mike Lee of Utah and Senator Mark Kirk of Illinois urged Trump to remove himself from the race.
The 2005 video showed Trump, then a reality TV star speaking on an open microphone about groping women and trying to seduce a married woman. The video was taped only months after Trump married his third wife, Melania.
Making a change at the top of the ticket for one of the two major parties this close to Election Day would be without precedent in the history of modern U.S. presidential elections.
In 1972, Democratic presidential candidate George McGovern's running mate, Thomas Eagleton, was replaced after it was revealed that the vice presidential hopeful had been hospitalized for depression. Eagleton withdrew from the race in August of that year and was replaced by Sargent Shriver.
In 2000, Missouri voters faced a last-minute candidate change in the state's U.S. Senate race after the Democratic challenger Mel Carnahan died in a plane crash, a month before the election. Carnahan's name remained on the ballot, however, and the dead man won the election. His widow, Jean Carnahan was appointed to fill the seat.
Republican party rules do not include a mechanism to involuntarily remove from the ticket a presidential candidate who has already been nominated.
The party does have procedure, known as Rule 9, that outlines the process for filling a vacancy if a nominee were to die or quit.
Republican leaders could try to pressure Trump into resigning by withdrawing financial support for his campaign and withholding logistical help. But as of Sunday, Trump showed no sign of backing down and went on the attack against party figures who have suggested he leave the race.
But if Trump were to decide to quit, the process would involve convening a 160-member committee of the Republican convention to select a new nominee, said Thomas Schwartz, a presidential historian at Vanderbilt University.
Experts say such a step would lead to electoral chaos with Republican attorneys fighting state-by-state legal battles to make the last-minute candidate swaps on paper ballots, long after many state deadlines have already passed.
Even more daunting, would be how to deal with early or overseas votes already cast. About 411,000 Americans already have voted in person or through the mail, according to Michael P. McDonald, a Florida State University political science professor who tracks early voting. The figure is sure to grow dramatically over the coming days. Early voting has already begun in six states and this week it will start in eight more.
Another problem for Republicans is the risk of legal challenges to the election result. Concern about such challenges could deter some Republicans from voting.
The Republican Party might argue that those ballots already cast for Trump should count toward electoral votes for the replacement candidate, said Matthew Dallek, an expert on presidential politics at George Washington University’s Graduate School of Political Management.
Democratic National Committee Chairwoman Donna Brazile said the party would likely challenge any effort by Republicans to replace Trump as their candidate on the ballot, adding that it would be "very confusing" for voters with early ballots already being cast.
"It would be political suicide for Donald Trump to step down right now," she told ABC News. "I'm sure my legal team is already looking at efforts underway, if there are any efforts underway," she said when asked if Democrats would oppose any effort by Republicans in the states to switch candidates on ballots.
Dallek said an effort to substitute a new candidate at this point would lead to "utter chaos."
"We would be operating in uncharted territory. You would be in this legal netherworld," he said.