“I give you my word as a Biden [that] I would not be running again if I didn’t believe with all my heart and soul I can do this job. Quite frankly, the stakes are too high.”

Nevertheless, Biden is facing calls from a number of Democratic strategists, donors and politicians to suspend his campaign and make way for a younger candidate at this year’s party convention in August.

One congressman told Matthew Yglesias, a US political blogger: “I think the president has one week to prove he is not dead.”

David Plouffe, who managed Barack Obama’s 2008 presidential campaign, told CNN: “It’s kind of a Defcon 1 moment…they are three years apart, but they seemed about 30 years apart tonight.”

David Axelrod, another Obama adviser, said: “There are going to be discussions about whether he should continue.”

Mark Buell, a well-known Democratic donor, said: “Do we have time to put somebody else in there?”

Few US elected politicians have declared publicly that Biden’s campaign should be ended, but dozens of Democrats have privately expressed serious concerns about his age and mental aptitude.

One lawmaker told NBC it was “time to talk about an open convention and a new Democratic nominee,” while Jim McGovern, another congressman, declined to publicly support the campaign and said he had not “come to any conclusion yet”.

Meanwhile, in the UK, Conservative Cabinet ministers privately called on the Democrats to ditch Biden in surprisingly frank terms in the context of the special relationship.

Three figures sitting around Rishi Sunak’s top table told The Telegraph that Mr Biden should be switched with someone better placed to defeat Donald Trump.

The sources variously called on the Democrats to “get rid” of Biden, to “switch him fast”, saying a younger alternative would have a much better chance of victory.

Calls for Biden to stand down began immediately after the end of the 90-minute debate in Atlanta, Georgia, which left supporters at an official viewing party with their heads in their hands.

The event went so badly that some speculated that Biden may have been encouraged to hold an unusually early debate to give his party time to find a replacement.

Several liberal media outlets that have generally dismissed concerns about Biden’s age published articles calling for a new nominee, while the president’s favourite news anchor, Joe Scarborough, said: “If he was CEO and he turned in a performance like that, would any Fortune 500 corporation in America keep him as CEO?”

On Friday night, The New York Times’s editorial board called for Biden to surrender the nomination to a “stronger opponent”.

The rules of the Democratic Party make it almost impossible for a challenger to remove Biden at this stage of the election cycle, after he won the party’s primary race virtually unopposed this spring.

But some in his party hope he will stand down voluntarily, and have discussed asking Obama, Hillary Clinton, or his wife Jill to urge him to do so.

It is thought that any attempt at a managed withdrawal from the race would also involve Chris Coons, a longtime Biden ally in the Senate, and Hakeem Jeffries, the minority leader of the House of Representatives.

Obama, one potential messenger, said Biden had had a “bad debate night” but that the election is “still a choice between someone who has fought for ordinary folks his entire life and someone who only cares about himself.”

Gavin Newsom and Gretchen Whitmer, the governors of California and Michigan, are under consideration as possible alternative nominees although Newsom said any talk of a replacement was “farcical”.

Biden’s allies rushed to his defence after the debate, pointing to a series of inaccurate statements by Trump and claiming the president had contracted a cold during his week-long preparation for the debate at Camp David.

Kamala Harris, the vice president, admitted his performance suffered from a “slow start,” but said he “got into a groove where it counted”.

Nancy Pelosi, the former House speaker and one of Biden’s closest allies, said the debate was his “worst night” but that his “presentation of integrity was far better than the other guy’s dishonesty”. She said she did not support the attempt to remove him as the nominee before November.

Biden’s campaign said he would not step down, and surprised some supporters by committing to taking part in a second debate against Trump on Sept 10.

“It wasn’t a good night,” said Jared Huffman, a Democratic representative, adding: “I don’t want him to do another debate.”

Opinion polls show that Biden and Trump are neck-and-neck nationally, but that the Republican candidate leads in all seven swing states. Without a significant swing towards him in those areas, Biden will lose the election.

In his campaign rally on Friday, the president referred repeatedly to “lies” from Trump on the debate stage and attacked him for his stance on abortion, plans for tax cuts for the wealthy, and support for Jan 6 rioters.

He gave a more energetic performance than on the debate stage, but stopped several times to cough as he read from an autocue.

“I know I’m not a young man,” he said. “I don’t walk as easy as I used to. I don’t speak as smoothly as I used to. I don’t debate as well as I used to.

“I know what I do know: I know how to tell the truth [and] I know how to do this job.”

Following the debate, the Trump campaign released a montage of Biden’s worst moments of the night.

Vivek Ramaswamy, a former Republican candidate and possible choice for Trump’s running mate, told The Telegraph: “I think he actually had a lobotomy beforehand, and I think what we saw is really a national security threat to the future of the country.”

Michael Whatley, the chairman of the Republican National Committee, said the campaign’s approach would not change “at all” if Biden was replaced.

“No Democrat is going to make a major change in policy, versus where Joe Biden is today,” he said.