As the race between Bernie Sanders and Hillary Clinton for the Democratic nomination comes down to the wire, the two candidates are looking ahead to their potential running mates — or at least ever-avid political reporters are. As the campaigns have been repeatedly prodded of late on their possible VP pick, one name is creating a lot of buzz: Elizabeth Warren.

When asked on MSNBC's Morning Joe on Tuesday if he would commit himself to nominating a woman as his vice president, Sanders demurred but said that he would give "very, very serious thought" to naming a woman as his running mate. Pressed on the matter, he named Warren, the progressive senator from Massachusetts, as a possibility.

"Elizabeth Warren, I think, has been a real champion in standing up for working families, taking on Wall Street," he said. "There are other fantastic women who have been active in all kinds of fights who I think would make great vice presidential candidates."

Sanders' statement comes just days after Clinton advisers, who are currently discussing possible vice presidential picks, also named Warren as a possibility in interviews with the New York Times.

For Clinton, the calculus is clear. Sanders has ignited the far-left wing of the party and brought out many new and independent voters who find Clinton's progressive bona fides lacking. A quarter of Sanders' supporters have said they would not vote for Clinton in the general election. But having a running mate with a strong, progressive resume like Warren's could help to bring some of Sanders' backers back into the fold.

Sanders, on the other hand, has a lot in common with Warren ideologically. They both built careers railing against a financial system that is unfair while promising to break up big banks and other power structures that supersede the needs of average Americans. Warren would be a partner in messaging, and could potentially help Sanders broaden his appeal to female voters.

But whether Warren would actually want the office is another question. John Nance Garner, Franklin Roosevelt's first vice president, is famously said to have remarked that the position isn't "worth a bucket of warm piss." Warren's office did not respond to a request for comment.

While both candidates have name-checked Warren, she has yet to do the same for either of them. She hasn't endorsed either candidate in the presidential race, and is just one of two senators to avoid wading into it. The former Harvard Law School professor, whose railing against big banks helped her to get elected to the Senate, where she is now one of its most prominent voices, has been conspicuous in her absence from the campaign trail this year.

In an interview with CBS last month, Warren said that she had no schedule for when she would dive into the Democratic race, but said that she was "glad" that both candidates were discussing major issues on the campaign trail that are important to her: money in politics, education, and inequality.

But Warren wasn't always silent on the 2016 race. Warren signed a letter in 2013 — along with every other woman in the US Senate — encouraging Clinton to run for president. The following year, as she denied her own interest in running for the presidency, Warren reiterated that Clinton would make a great president.

"I hope she does. Hillary is terrific," she told ABC's This Week at the time.

But that was long before Sanders jumped into the race and proved to be a significant challenger. Since, Warren has stayed mum — at least on the Democratic side.

Clinton's advisers told the New York Times that she is less concerned about finding a running mate whose personality or ideology will help her campaign than she is about finding someone who can dominate the vice presidential debates and be "an effective attack dog" against the Republican nominee.

Though Warren has been careful to avoid taking sides in the Democratic primary, she hasn't shied away from attacking Republican candidates.

Last month, Warren laid out a viral social media assault against Donald Trump, calling him a "loser" and a "wannabe tyrant" who could be "a serious threat" while comparing him to "history's worst authoritarians" in a series of tweets.

Last week, Warren started up a new tirade against Senator Ted Cruz, responding to a fundraising email that the Texas senator sent out to supporters highlighting how running for president is "a significant sacrifice," describing how he doesn't get to see his family often, loses sleep, and must withstand constant attacks from his opponents.

It's unclear whether Warren will weigh in on the Clinton-Sanders battle before the Democratic convention in July. The two candidates will face off in five state primaries Tuesday night, as their race for the nomination nears its final stages.