Harris and her advisers did not take up the offer to meet and have shown no interest in the proposal, which Kennedy called a “strategic mistake”.
He told the Washington Post: “I think they ought to be looking at every opportunity. I think it is going to be a very close race.”
Kennedy also attempted to contact Biden via his former White House chief of staff and longtime adviser, Ron Klain, according to the newspaper.
It comes after Kennedy made similar overtures to Donald Trump last month at a meeting at the Republican national convention in Milwaukee.
Kennedy also recorded a phone call between himself and the Republican candidate in which Trump said: “I would love you to do something - and I think it would be so good for you and so big for you”.
The Kennedy campaign claims to have polling demonstrating that Harris or Trump would receive a boost by announcing Kennedy could have a Cabinet role in their administration.
Despite his outreach to both main parties, Kennedy told the Washington Post he intended to continue his long-shot White House bid.
However, his advisers said he was open to withdrawing if he finds another way to bring about the change he seeks in the country.
Stefanie Spear, a spokeswoman for the Kennedy campaign, told CNN that Kennedy was “willing to meet with leaders of both parties to discuss the possibility of a unity Government”.
Democrats have cast Kennedy as a “spoiler” candidate who could hand victory to Trump, and highlighted his support from the Trump-backing Republican mega-donor Timothy Mellon, to argue the point.
“No one has any intention of negotiating with a MAGA-funded fringe candidate who has sought out a job with Donald Trump in exchange for an endorsement,” said Matt Corridoni, a spokesman for the Democratic National Committee.
