Mr Musk’s donation comes as President Biden’s funding has taken a hit following his catastrophic debate performance against Trump.
Key Democrat donors including George Clooney, the actor and film-maker, have publicly called for the president to step aside amid concerns about his mental acuity.
On Friday night, Mr Biden held a rally in the swing state of Michigan in which he accused journalists of “hammering” him.
The 81-year-old walked on stage in Detroit to chants of “don’t you quit”, and insisted that he will remain in the presidential race despite growing calls from within his party to drop out.
He also addressed reporting of his gaffes on the world stage at the Nato conference, where he introduced Volodymyr Zelensky as “President Putin”.
Mr Biden acknowledged he had been panned by critics in the media, but suggested that the same scrutiny had not been applied to his Republican rival.
“Now I’m not complaining, I’m just saying,” Mr Biden began.
“You may have noticed that since the debate, the press, and they’re good guys and women up there, they’ve been hammering me. I make a lot of mistakes.”
As he spoke, the crowd began booing and jeering at reporters who were assembled to cover the event.
The president protested at the crowd’s reaction, telling supporters: “Oh no, no, no, it’s okay.
“They’ve been hammering me, because I sometimes confuse names. I say that’s Charlie, instead of Bill.
“But guess what, Donald Trump has gotten a free pass.”
The incident was compared online to Trump’s tactic of referring to negative press coverage while speaking at rallies, after which attendees have jeered at the press.
My health a legitimate concern, says Biden
Mr Biden had earlier acknowledged his health is a “legitimate concern” for voters but doubled down on his insistence he would not be leaving the presidential race.
Twenty-one sitting Democrats have publicly called for him to stand down, and senior party figures including Barack Obama and Nancy Pelosi are reported to have concerns about his leadership.
Among those to urge Mr Biden to step aside is California Democrat Mike Levin, who addressed Mr Biden directly in a virtual meeting with members of the Congressional Hispanic Caucus.
During the meeting, the Democrat leader noted that his achievements were not cutting through with voters like they did before.
‘“That was great when you were feeling good, Biden – are you OK now?’ That’s what’s underlying. That’s what people are worried about,” he told Mr Levin, the New York Times reported.
“It’s a legitimate concern for people, but that’s why I think it’s important – I’ve got to get out and show people everything from how well I move to how much I know and that I’m still in good charge.”
Mr Biden said he was “going out and letting people touch me, poke me, ask me questions” in order to quell concerns.