The Minnesota senator was speaking at a campaign rally for Mr Biden on Saturday in Michigan when she sparked the rumours.
"I could not think of a better way to end my candidacy, as hard as it was to do with our beloved staff and everyone else, than to join the tick–" she's heard saying to the crowd.
It sounded like Ms Klobuchar wanted to say the word "ticket", hinting she could join Mr Biden as his vice president pick.
People in the Grand Rapids crowd thought the same thing, and they started cheering and chanting the senator's name.
Ms Klobuchar then corrected herself by saying she was going to say "terrific" not "ticket".
"Guys, I was going to say ... ," she said over chants of her name, "than to join the terrific, the terrific, terrific campaign of Joe Biden."
Ahead of Super Tuesday, the senator dropped out of the race to earn the Democratic nomination and endorsed the vice president shortly after. Her endorsement helped boost him in the polls and come away as the frontrunner ahead of Bernie Sanders.
Mr Biden has not revealed who he will select to be his running mate, but Ms Klobuchar's name has been tossed around as a potential option for him. Senator Kamala Harris' name has also been floated around after she dropped out of the race. She endorsed his run in a statement on Sunday morning.
Last fall, the vice president mentioned several possibilities, including former Georgia governor candidate Stacey Abrams and former Deputy Attorney General Sally Yates. But whether he's asked anyone to officially join his ticket remains unknown.
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