Mr Biden, a former vice-president, told reporters he spoke to Ms Harris on Tuesday and believes she is a strong potential running mate.
“Look, Senator Harris has the capacity to be anything she wants to be,” he said.
“She’s solid, she can be president someday herself, she can be vice president, she could go on to be a Supreme Court justice, she could be attorney general. I mean she has enormous capability.”
Ms Warren told NBC she believes the California senator would be on “any Democrats shortlist” for vice-president.
Ms Harris, who was seen as a potential frontrunner for the Democratic nomination, said she was ending her campaign due to funding issues earlier this week.
“I’m not a billionaire. I can’t fund my own campaign. And as the campaign has gone on, it’s become harder and harder to raise the money we need to compete,” she wrote in a statement.
The senator notably clashed with Mr Biden during the first Democratic debates and challenged him on his record on racial issues, specifically when he opposed federally-mandated busing to desegregated schools.
The confrontation was widely seen as the standout moment from the debate and led to a brief bump in her polling numbers.
At the start of the second debate, Mr Biden was caught on camera asking her to “go easy on me, kid” in an apparent reference to their earlier tense exchanges.
“She was down in the polls, her fundraising dried up and she didn’t want to be on the California ballot and come in fifth or sixth,” Mr Shrum said.
“All that said, she’ll be top-of-the-list for vice president if Biden, Buttigieg or even Sanders is the nominee.”
Voting for the nomination process will start with the Iowa caucus on 3 February, 2020.