The White House has fought back against questions about Joe Biden’s mental fitness, with the US president telling anxious fellow Democrats it is time to end the criticism, and daring doubters in the party to challenge him.

In a tense news conference, the president’s spokeswoman rejected suggestions that he might be suffering from an undisclosed illness.

Mr Biden, 81, himself took the unusual step of calling in to MSNBC’s morning show, saying: “I am not going anywhere.”

Questions about his mental acuity have intensified since a poor debate performance against Donald Trump on 27 June.

The scrutiny is unlikely to fade this week as he hosts a summit in Washington for leaders of Nato countries.

In Monday’s daily press conference, White House press secretary Karine Jean-Pierre rejected speculation that Mr Biden was being treated for Parkinson’s disease, which can cause stiff movement and slurred speech.

“Has the president been treated for Parkinson’s?” she said. “No. Is he being treated for Parkinson’s? No.”

An expert on Parkinson’s disease from Walter Reed National Military Medical Center near Washington DC has visited the White House eight times since last year, the New York Times reports.

But White House records show the same doctor also made a number of visits during the Obama administration from 2012-16.

Ms Jean-Pierre would not comment on the matter, citing the need to protect the privacy of the doctor and security reasons.

On Monday, the president called in to left-leaning MSNBC’s Morning Joe programme, laying down the gauntlet to critics to “challenge me at the convention” next month, or rally behind him against Trump.

In a letter sent to congressional Democrats also on Monday, Mr Biden said he “wouldn’t be running again if I did not absolutely believe” that he could beat Trump, the Republican challenger in November’s election.

Mr Biden’s letter said he had “heard the concerns that people have” and “is not blind to them”, but that Democratic voters in the primaries have “spoken clearly and decisively” that he should be the party’s nominee.

“Do we now just say this process didn’t matter?” the letter said. “That the voters don’t have a say… I decline to do that. How can we stand for democracy in our nation if we ignore it in our own party? I cannot do that. I will not do that.”

Mr Biden also phoned Democratic donors on Monday. One source familiar with the call told CBS News, the BBC’s US partner, that Mr Biden said his strategy for the second debate against Trump in September will be “attack, attack, attack”.

It comes a day after the Democratic minority leader in the House of Representatives, Hakeem Jeffries, held a group call in which four congressmen were explicit in urging Mr Biden to step aside, according to US news outlets.

The quartet was joined by others who voiced concerns about Mr Biden’s fitness for office after his stumbling debate performance, but stopped short of asking for the president to exit the race.

Jerry Nadler of New York, Mark Takano of California and Adam Smith of Washington state all said Mr Biden should step aside, according to multiple outlets, citing people on the call or those familiar with what was said.

Joe Morelle of New York added his voice, according to CBS and the New York Times, but the Associated Press said the fourth person was Jim Himes.

Others on the call expressed concern about Mr Biden’s electoral chances against Trump.

On Monday, Mr Smith publicly called for Mr Biden to quit, saying in a statement “the American people have made it clear they no longer see him as a credible candidate to serve four more years as president”.

But another Democratic member of Congress, Alexandria Ocasio-Cortez, spoke up on Monday in support of Mr Biden.

“The matter is closed,” the left-wing New York lawmaker told reporters. “Joe Biden is our nominee. He is not leaving this race. He is in this race and I support him.”

Last week, Lloyd Doggett of Texas became the first Democrat in Congress to urge Mr Biden to step aside.

Trump, 78, has ridiculed Mr Biden over the debate, last week labelling his rival “broken-down”. Biden allies have expressed exasperation about the media criticism he is facing, while his Republican challenger was recently convicted in a New York hush-money case.

Amid mounting speculation over Mr Biden’s candidacy in November, the thoughts of some Democrats have turned to who could replace him.

Some party members have rallied around Vice-President Kamala Harris, who is Mr Biden’s running mate in November.

Trump has suggested the vice-president would be “better” than Mr Biden, but still “pathetic”.

During a pair of interviews last week, Mr Biden acknowledged that he had “screwed up” the debate, but later vowed that only the “Lord Almighty” could convince him to end his bid to win the White House again.