It has long been known that many in the Republican Party are deeply concerned about their candidate’s performance and behaviour, and would love even at this late stage to get rid of him.

Yet the furore, and the ongoing bad headlines, have led some to speculate that the 70-year-old New York tycoon may simply to decide to pack it all in on own accord.

If he were to do that, it would be the first time a presidential candidate had dropped out, but there have been instances involving vice-presidential candidates leaving the race. ABC News reported that senior party officials were looking into options for the 8 November election if Mr Trump decided he had already had enough.

For what it is worth, his campaign spokesperson, Hope Hicks, dispatched an email telling reporters: “There is no truth to this whatsoever.”

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Is Trump losing it?

A flurry of articles have appeared in recent days questioning the sanity of the Republican candidate. These spiked after Mr Trump called on Russia to hack into Hillary Clinton’s emails, a seemingly treasonous comment that he later said was merely a joke. On MSNBC, Joe Scarborough said he had been taking calls from people inside the party asking questions about Mr Trump’s well-being.

“I fielded calls all day yesterday, from conservatives, from Republicans, from officials, people that the media would call right-wing bloggers, and everybody was asking about his mental health,” he said. The Washington Post’s Eugene Robinson wrote a column entitled: “Is Donald Trump just plain crazy?”

President Barack Obama added to the debate, saying he believed the tycoon was "woefully" unprepared for the demands of office.

On Wednesday, Mr Trump sought to dispel suggestions that anything was wrong by taking to Twitter to say: “There is great unity in my campaign, perhaps greater than ever before. I want to thank everyone for your tremendous support. Beat Crooked H!”

What would happen if he pulled out?

Essentially, it would fall to the Republican Party to scramble like crazy to find a replacement. Robert Shapiro, professor of government at Columbia University, told The Independent the scenario was covered by the party’s rules and by-laws.

Reports suggest Rule 9 of the Republican National Committee (RNC) relating to “Filling Vacancies in Nominations”, stipulates that should the party’s presidential or vice-presidential candidate leave the ticket, the slot may be filled either by a reconvening of the national convention or by the party committee itself.

If the party decides not to have another convention, the Republican National Committee (RNC) members representing a given state are entitled to cast the same number of votes as that state was entitled to at the convention.

Is there a precedent for this?

“I’m sure you know this is just a rumour and apparently has no foundation,” said Larry Sabato, director of the Centre for Politics at the University of Virginia. “If it somehow happened, the RNC alone would substitute a nominee. No new convention would be called.”

He added: “There are two historical precedents, both involving the vice-presidency. In October 1912, President William Howard Taft’s vice-president died in office, just days before the election. The RNC replaced him. It didn't matter since Taft was overwhelmingly defeated for re-election by Democrat Woodrow Wilson.

“In 1972 the Democratic VP-nominee, Senator Thomas Eagleton of Missouri, dropped off the ticket just weeks after having been nominated, because it was discovered that he had had serious mental health problems requiring electric shock therapy. He was replaced by the DNC; JFK brother-in-law Sargent Shriver was chosen by presidential candidate George McGovern, and ratified by the DNC.”

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Who would replace him?

Mr Shapiro said that the RNC could replace Mr Trump with any candidate of its choice, and would not require that person to have run in the primary. It could be Mike Pence, the VP candidate, it could be Ted Cruz, it could be John McCain.

He said that there would probably be a lot of pressure from within the Republican establishment to nominate a “uniting candidate”, such as Paul Ryan, the Speaker of the House, who has been careful to keep his distance from Mr Trump. He said that as the last two candidates in the primary race against Mr Trump, Mr Cruz and Ohio Governor John Kasich would likely make a vocal case for their candidacies.

What if Trump decided to run as an independent?

There is speculation that Mr Trump might chose to do this simply to annoy the RNC. If he were to do this, it would almost certainly split the non-Democratic vote and hand the election to Ms Clinton. There are strict rules, different in each state, governing when a candidate puts their name on the ballot. If he missed the deadlines, the only alternative would be to run as a so-called “write-in candidate” where his name would not appear on the ballot and voters would choose to support him by writing his name on the form.

Would it delay the election on 8 November?

Almost certainly not.

Just how likely is it?