Mr Biden and his campaign team have ruled out leaving the race against Trump, arguing that he is the only candidate that could win an election against him and that it is too late to change the party’s nominee.
But Ms Pelosi, who is one of the 81-year-old’s closest political allies, said she and others had “made concerns known” to the campaign team since Thursday’s debate.
“I think it’s a legitimate question to say, ‘Is this an episode or is this a condition?’” she said.
“And so when people ask that question, it’s legitimate – of both candidates.”
She added: “Both candidates owe whatever tests you want to put them to, in terms of their mental acuity and their health.”
The 84-year-old California representative said she had not spoken to Mr Biden since the debate, but had spoken to members of his team.
“We all have been in touch with people close to the president, so it’s not a question of not having an opportunity to make our concerns known or have questions answered,” she said.
She also defended Mr Biden’s record during the coronavirus pandemic, on veterans’ issues and economic policy.
Karine Jean-Pierre, the White House press secretary, denied that Mr Biden had dementia or a degenerative disease.
She added: “He was under the weather obviously. We all have bad nights, bad moments, it is not unusual.
“For most incumbents, the first debate doesn’t go well. We are going to move forward.”
Mr Biden would be doing a television interview as well as visiting Wisconsin and Pennsylvania, she said, adding: “We want the turn the page on this. We want to turn the page for the American people.”
On Wednesday, Mr Biden will hold a call with Democrat governors to discuss his campaign, following days of speculation that he may leave the race.
The discussion will include Gavin Newsom, Gretchen Whitmer and JB Pritzker – three governors who have been touted as possible replacements.
It follows a separate call on Monday organised by Tim Walz, the governor of Minnesota, in which Democrats said they would like to speak to the US president about his campaign.
A source on that call told NBC News that the governors were not calling for Mr Biden to stand back, but questioned the campaign’s “strategy” after the debate.
“What everybody needs is reassurance,” they said. “No one said we need to go another direction.”
Mike Quigley, another Democrat from Illinois, said Mr Biden should be “honest” about the impact of the debate.
“It wasn’t just a horrible night,” he said. “I just want him to appreciate at this time just how much this impacts not just his race but all the other races coming in November.”