Mr Biden could be replaced by Kamala Harris, his vice-president, although there are concerns she would prove even less popular than he is.
Other options include Gavin Newsom, the charismatic governor of California, although he remains supportive of the president staying in the race.
Experts say it would be virtually impossible for the Democrats to oust Mr Biden at this late stage of the campaign, and that in reality the decision lies solely with the president.
Mr Biden’s calamitous performance at the debate has rattled top donors to the Democrats, with one reportedly asking the campaign team directly whether the president was going to drop out soon.
“They’re [Biden’ team] saying, ‘we just had one bad night…what they’re missing, a vital point they’re missing, is it’s not just one bad night. There’s no fixing this,” one Democrat donor told Politico, a US news site.
There is also concern that some Democrat party donors may pull funding if they feel there is no hope of Mr Biden recovering from the debate. “The biggest problem is donors. If the money dries up, everybody will start to panic,” another source said.
Some Democrats have openly poured scorn on their leader, with Jay Surdukowski, an activist and lawyer from New Hampshire, declaring: “Biden is toast – calling it now.”