In an interview with GQ published last month, Biden reiterated that he could make a bid for the Oval Office.

"I can die a happy man never having been president of the United States of America," he said. "But it doesn't mean I won't run."

He also hinted that Clinton's presence in the race wouldn't necessarily deter him.

"The judgment I'll make is, first of all, am I still as full of as much energy as I have now — do I feel this?" he said. "Number two, do I think I'm the best person in the position to move the ball? And, you know, we'll see where the hell I am."

Still, he'd be fighting an uphill battle if he did opt to run. A McClatchy-Marist poll released late last month found that among Democrats nationally, 63 percent would favor Clinton as their party's nominee, versus 13 percent who would back Biden. No other candidate broke double digits, while 18 percent remained undecided.

Clinton, for her part, hasn't decided whether she'll run. But her post-retirement schedule, which includes a cross-country blitz of speaking engagements, has many suggesting that she will seek the nomination.

Last week, New York City mayoral candidate Anthony Weiner hinted that his wife, top Clinton aide Huma Abedin, knew what her role would be in the Clinton campaign — seemingly revealing that the effort was already under development. Weiner later said the comment was a joke, and a spokesman for Clinton dismissed the comment.