Biden, 81, said any candidates who doubt his ability should challenge him at the Democratic National Convention in August - an effort that stands no chance of success unless he lets the delegates he won in primaries this year consider other candidates.
"The bottom line here is that I am not going anywhere," Biden said in a phone call he placed to MSNBC's Morning Joe programme. He repeated that message to donors on a private call later in the day, according to two sources on the call.
Biden faces a critical week as he tries to shore up a campaign that has been on defence since a shaky June 27 debate against Republican Donald Trump in Atlanta, which raised questions about his ability to do the job for another four-and-a-half years.
Several congressional Democrats have called for him to drop out.
Senator Dick Durbin, the chamber's No 2 Democrat, told reporters Biden's candidacy "will be thoroughly discussed this week, as it should." He said he had spoken with about a dozen of his colleagues, who held a range of views.
A growing number of Democratic lawmakers have voiced concern that Biden's poor public approval ratings, plus worries about his age and ability, could hurt their prospects for retaining the Senate, which they control by a 51-49 majority, and winning back the House, where Republicans have a 219-213 majority.
"I think we need a different choice if we're going to be able to beat Donald Trump," Democratic Representative Adam Smith said on CNN.
Another Democrat, Representative Joe Morelle, said many of his constituents had told him they worried Biden was not capable of serving as president, though he stopped short of calling on Biden to drop out.
"The bigger question is, how does he dispel concerns that were raised from his performance in the debate?" Morelle told reporters.
Not being treated for Parkinson's - White House
Concerns that Biden might be suffering from an undisclosed illness have risen since he stumbled, appearing frail and losing his train of thought at times at his debate against Trump.
His halting debate performance has prompted some to question whether he was suffering from a neurological disorder such as Parkinson's disease, which can cause slurred speech and halting movement.
But White House spokesperson Karine Jean-Pierre said Biden was evaluated and given a clean bill of health in February and is not being treated for the disease.
The New York Times has reported visitor logs showed Dr Kevin Cannard, a specialist in Parkinson's from Walter Reed National Military Medical Center, had visited the White House at least eight times from last August through March.
Dr Kevin O'Connor, the White House doctor, issued a letter on Monday night that said Biden has not seen a neurologist outside of his normal annual physical.
O'Connor said Cannard's visits to the White House were part of his neurological clinics aimed at supporting "thousands" of active duty members assigned in support of White House operations and not to treat the president.
Cannard's visits to the White House were first reported by The New York Post on July 6.
One former White House official, who worked for Biden when he was vice president under President Barack Obama and as president, was treated by the same doctor for migraines in 2016, the former official told Reuters.
The doctor came to the White House once a month then as part of his rotation, the individual said.
White House spokesperson Karine Jean-Pierre refused to confirm or elaborate on Cannard's visits earlier in the day. At the media briefing, she was frequently challenged by reporters and said she wanted to respect the privacy of all involved for security reasons.
She said Biden had seen a neurologist three times connected to his annual physical exam. She did not explain Cannard's presence at the White House but suggested it might be connected to treating some of the military personnel who work at the White House complex.
"There are thousands of military personnel that come to the White House and they are under the care of the medical unit," she said.
'Biden is the nominee'
If Republicans were to capture the White House and both houses of Congress, Trump, 78, would face few constraints on his ability to push through major policy changes.
Some Democrats expressed concern but stopped short of calling on Biden to drop out of the race.
"President Biden has got to prove to the American people - including me - that he's up to the job for another four years," said Senator Jon Tester, who faces a tough re-election fight in Montana.
Other lawmakers said they supported his candidacy.
"President Joe Biden is the nominee and has been selected by millions of voters across this country," Representative Steven Horsford, chair of the Congressional Black Caucus, said on social media. Black voters are a critical part of Democrats' base of support.
In his letter to Democrats, Biden said he was aware of their concerns but said it was time to put them aside.
Biden will have little time to campaign this week as he hosts a meeting of NATO member states, capped with a rare solo news conference on Thursday.
A Reuters/Ipsos poll last week found that one in three registered Democratic voters believed that Biden should quit the race, with 59% of respondents in the president's party saying he is too old to work in government.
However, that poll also found that none of his possible replacements fared better in a matchup against Trump. The poll found Biden and Trump tied at 40% each.
Trump predicts Biden will stay in race
Trump said he expected Biden to remain in the race for the White House despite growing calls within the Democratic Party for him to step aside for a younger nominee.
Trump said on Monday that Biden would refuse to quit the campaign because of his ego and noted that the president had the Democratic nomination locked up because of all the convention delegates he won during the Democratic primary elections earlier this year.
Biden has repeatedly said he is staying in the race.
"Interestingly, he's got a lot of power, because he's got the delegates. You know, when you have the delegates, unless he says, 'I'm getting out,' they can't do anything to get him out other than the 25th amendment," Trump told interviewer Sean Hannity in a telephone call broadcast on Fox News.
The 25th amendment of the US constitution enables the vice president and cabinet members to declare the president is unable to discharge the powers and duties of the office, transferring the job to the vice president as acting president. There is no indication Vice President Kamala Harris or top Democrats have advanced that option.
"He's got an ego, and he doesn't want to quit. He doesn't want to do that. It just looks to me like that's what he wants," Trump said of his rival, who unseated Trump when the former president ran for re-election in 2020.
Trump's interview contradicted casual statements captured on video recently, when he falsely told supporters at one of his golf courses that he had driven Biden out of the race.
"He's quitting the race ... I got him out," Trump says in a video published by the Daily Beast last week.
