However, Mr Biden is facing growing pressure from party insiders to keep going because Ms Harris is floundering in polls.
This week it emerged Ms Harris has been consulting other high profile female politicians, including Angela Merkel and Hillary Clinton.
Mr Biden is already the oldest person ever to be inaugurated as US president.
If he were to win the next election he would be 86 by the time he leaves office.
Ms Harris said last week that she has not discussed with Mr Biden whether he will run in 2024.
Failure in the face of omicron
Meanwhile, Mr Biden has been accused of failing to provide coronavirus tests quickly enough to tackle the omicron variant.
This week he announced he would send 500 million free test kits to Americans’ homes.
But a contract to buy the tests has yet to be signed and they first will not be available until January.
Mr Biden admitted that “nothing’s been good enough” when it came to the availability of tests.
The president said he “wished I had thought about ordering” the 500 million tests “two months ago.”
Meanwhile, Mr Trump took credit for the vaccines and urged people to get vaccinated.
He said: “I came up with a vaccine, with three vaccines. Look, the results of the vaccine are very good, and if you do get it [the virus], it’s a very minor form.
“People aren’t dying when they take the vaccine.”