During a fundraiser alongside Mr Obama on June 15, it was widely reported that Mr Biden “appeared to freeze up” on stage.
In clips of the incident, Mr Biden pauses in front of the crowd and raises his hand to take in the applause from the audience.
After dropping his arm to his side, he then stands stiffly on stage before Mr Obama tugs his wrist to walk offstage together.
What happens now?
It would be challenging for the Democratic Party to replace Mr Biden at this stage in the campaign, after he won the party’s primary essentially unchallenged earlier this year.
Incumbent presidents running for a second term are almost never defenestrated by their party at their party’s convention, where the formal decision to award a nomination takes place.
To replace Mr Biden, the party would either require his consent, or for more than half of the state delegates who pledged support for him during the primary process to turn their backs on him.
If Mr Biden agreed to step aside, then the party would choose a new nominee in an open process of voting on the convention floor in Chicago in late August.
Politico reported on Thursday night that a strategist close to potential Democratic presidential candidates had been bombarded with messages requesting that their boss step up to challenge Mr Biden.
Kamala Harris, his vice-president who is reportedly on the shortlist to replace him, spoke in support of Mr Biden.
She told CNN that Mr Biden had offered “a very clear contrast with Donald Trump on all the issues that matter to the American people”.
“Yes, there was a slow start, but it was a strong finish,” she said.