Despite this, the White House and the Biden campaign have vehemently denied reports he is actively weighing his future and say he is committed to defeating Trump for a second time on 5 November.
The New York Times and CNN reported on Wednesday that Mr Biden had told an unnamed ally he was evaluating whether to stay in the race.
Both reports said the president had told the ally he was aware his re-election bid was in danger and his forthcoming appearances – including an ABC News interview and a Friday rally in Wisconsin – were hugely important to his campaign.
A spokesperson rejected the reports as “absolutely false”, shortly before White House Press Secretary Karine Jean-Pierre faced a barrage of questions about Mr Biden’s commitment to the race.
She said the reports he may drop out were untrue: “We asked the president [and] the president responded directly… and said ‘no, it is absolutely false’. That’s coming direct from him.”
Mr Biden met 20 Democratic governors from around the country, including California’s Gavin Newsom and Michigan’s Gretchen Whitmer, later on Wednesday. Both have been tipped as potential replacements if Mr Biden were to stand aside.
“The president has always had our backs, we’re going to have his back as well,” Maryland Governor Wes Moore told reporters after the meeting.
New York Governor Kathy Hochul said the two dozen governors who had just met the president pledged their support and that Mr Biden had vowed he was “in it to win it”.
But Ms Harris is still considered the most likely replacement. The 59-year-old has been hampered by poor approval ratings, but her support has increased among Democrats since the Biden-Trump debate.
The vice-president gave an immediate interview on CNN after the debate, projecting calm as she expressed full support for the president.
“She’s changing nothing,” a source close to Ms Harris told BBC News, adding that she would continue to hit the road on behalf of the campaign.
“She has always been mindful to be a good partner to the president,” said Jamal Simmons, Ms Harris’ former communications director.
“The people who ultimately will make the decision about who the nominee should be mostly are people who are pledged to him. Her best role is to be a partner to him.”
Members of the Democratic National Committee are charged with voting to officially make President Biden the party’s nominee at the August convention, putting him on the ballot nationwide.
One member, who has spoken to other delegates and requested anonymity to speak frankly about sensitive discussions, told the BBC that the nomination should go to Vice-President Harris if Mr Biden opted not to run.
“If we open up the convention, it will cause pure chaos that will hurt us in November,” they said.
A report by the Washington Post, meanwhile, said Mr Biden and his team recognised that he must demonstrate his fitness for office in the coming days.
He appeared at a Medal of Honor ceremony on Wednesday, and has planned trips to Wisconsin and Philadelphia later in the week.