Politico on Thursday night cited an unidentified source as saying Clinton and her aides were discussing how to approach Biden about the post, should she win the November 8 presidential election.
"We need your muscle, we need to move", Biden told the group.
Clinton and Biden have had a lengthy rivalry despite both having served in Obama's cabinet.
"If I don't move, I'll be demoted to secretary of state or something", he said, as a slow wave of laughter rose from the surprised crowd. Before joining the Obama administration, Biden chaired the Senate Foreign Relations Committee.
Just on Monday, on a stop at a Clinton campaign office in Toledo, Biden said that he'd spoken to the Latvian president, who'd urged him to come to Europe and reassure people that if Russian Federation invades, North Atlantic Treaty Organisation will defend them. There may be some lingering issues from Biden's potential presidential run, and he's said to be sensitive to the suggestion that he opposed the raid that killed Osama bin Laden while Clinton was strongly in favor of it. President Obama announced in January that Biden would head up the efforts, after he lost his 46-year-old son Beau Biden to brain cancer in May of 2015.
Politico first reported Mr Biden was "at the top of the internal short list" of the Clinton team who are planning for her winning the White House.
Clinton insists publicly she is focused exclusively on winning the election in 11 days time. Biden has said he will spend the rest of his life working on that project, in some form.
The vice president said at a campaign event for Clinton in Pennsylvania this week that he is frequently asked whether he wished he had been debating Trump, who has insulted women, Muslims and immigrants and thumbed his nose at political norms.
While he is more than qualified, and has cross-party appeal, Clinton would also be selecting someone she has had major clashes with in the past - including whether or not to leave troops in Iraq or to bomb Syria.
Hillary Clinton and Joe Biden on August 15 in Scranton, Pennsylvania.
If Biden is picked to be Clinton's Secretary of State, he clearly seems up to the job as of now, as he travels across the country campaigning for Hillary.