Women have been tapped twice before as vice-presidential candidates, but with polls showing Biden leading President Trump in a general election, many see this as the most realistic possibility that a woman could wind up a heartbeat from the presidency.

Biden, 77, has described himself as a “bridge” to the next generation of leaders, a comment widely interpreted as a signal that he’d serve just one term, meaning his running mate would be even more of a president-in-waiting than usual.

“There is this invisible glass ceiling that we just saw play itself out in even the Democratic nomination process,” said Randi Weingarten, president of the American Federation of Teachers and a major player in the party. “We have to find a way to change this construct of there being this invisible bar to women becoming president or vice president of the United States.”

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A person close to the Biden team who was not authorized to speak publicly, and so spoke on the condition of anonymity, said the campaign will soon announce a running mate vetting operation, but warned that the campaign is in the midst of growing quickly after its rocky start.

Biden has told MSNBC that it’s “very important” to pick someone who’s been tested via a presidential campaign, suggesting that the women who ran against him may have an edge. Sens. Amy Klobuchar (D-Minn.) and Kamala D. Harris (D-Calif.) have both endorsed Biden, and Klobuchar in particular has gone out of her way to show her loyalty.

The senator from Minnesota heavily retooled her own campaign in her home state to support Biden and appeared at two rallies in Michigan on his behalf. She noted on CNN on Monday that Biden then won those two states, which will be crucial in the general election, adding that it’s “a great thing” that Biden is planning to select a woman.

Sen. Elizabeth Warren (D-Mass.) has not endorsed Biden, but the two have spoken twice since she dropped out of the race this month. Their most recent conversation included a discussion of her plan to overhaul the country’s bankruptcy system, which he adopted this week and touted on the debate stage Sunday.

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The move was widely seen by Warren allies as an olive branch to the Bay State senator, who famously clashed with Biden in the 1990s on bankruptcy laws. A person close to Biden, who was not authorized to discuss campaign strategy, said Biden had been interested in Warren’s plan since she unveiled it in early January.

Biden has previously hinted that he’d be interested in running with Warren, floating the idea of a joint ticket in 2015 as he was considering a challenge to Clinton. In the end he chose not to run.

The former vice president has “strongly believed” for “some time” that his running mate should be a woman, according to a Biden adviser who was not authorized to speak publicly about his selection. The adviser added that Biden “has been eager to make this public once it was appropriate based on his standing in the race.”

The disappointment of many women with the outcome of the Democratic primary contest is palpable, especially after Clinton’s devastating loss in 2016. The initial field of more than two dozen hopefuls included several prominent women, including four U.S. senators, but Harris and Sen. Kirsten Gillibrand (D-N.Y.) never got traction, and neither Warren nor Klobuchar finished higher than third in any state.

Biden’s move could be a way to reach out to a pivotal voting bloc. His announcement — made Sunday in response to a question during the first Democratic debate since 2004 that did not include a woman — comes at a moment of reckoning for a party that has long touted a commitment to diversity.

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Women have led the opposition to Trump, first by holding massive marches in cities around the globe the day after his inauguration, and then by running for office in record numbers and winning a historic number of seats in Congress, which elevated Nancy Pelosi to be House speaker.

So far, 56 percent of Democratic voters this year have been women, according to a Washington Post analysis of entrance and exit polls. That’s in line with 2016, when 58 percent of the Democratic primary electorate was women, and 2008, when women made up 57 percent of the vote.

During Sunday’s debate, Biden and Sen. Bernie Sanders (I-Vt.) were asked how they would select Cabinet members able to advise them on “issues that affect women’s physical and financial health.”

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“I’ll pick a woman to be vice president,” said Biden, who was apparently waiting for an opportunity to make this announcement. “There are a number of women who are qualified to be president tomorrow. I would pick a woman to be my vice president.”

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He also repeated a pledge to appoint the first black woman to the Supreme Court.

Sanders committed to appointing women to at least half the top administration jobs and, when pressed, said he would “in all likelihood” select a woman to be his running mate.

On the campaign trail, Biden at various points has mentioned a number of women as suitable running mates, beyond his primary opponents. They include Stacey Abrams, the 2018 Democratic gubernatorial nominee in Georgia; and Sally Yates, the former acting U.S. attorney general who refused to carry out Trump’s order to close the U.S. border to people from seven majority-Muslim countries.

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Others in the party have mentioned Michigan Gov. Gretchen Whitmer, whose state will be critical in the general election, and Sen. Catherine Cortez Masto (D-Nev.), a prominent Latina from another important region.

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The United States lags far behind other democracies, from Britain to Germany to Israel, in putting a woman in the top job. If Biden wins and a woman becomes vice president, it could acclimate voters to the image of a woman at the pinnacle of power.

At the same time, the number of women in the primary field, and the strong performance of Warren and others on the debate stage, added pressure on Biden to make his announcement.

“Now that we’ve seen what a diverse debate stage can look like, it’s hard to imagine being satisfied with anything else,” said Barbara Lee, a Massachusetts-based philanthropist whose foundation focuses on electing women to executive positions. “I believe that a more inclusive ticket is essential, to energize and mobilize 2020 voters.”

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She said the change could be permanent. “Moving forward, it’s hard to imagine a presidential cycle without a woman on the ticket,” Lee said.

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Last week leaders from nearly two dozen women’s groups and liberal organizations — including Emily’s List, Planned Parenthood Action Fund, UltraViolet and the Sunrise Movement — sent a letter to the Biden and Sanders campaigns, as well as the Democratic National Committee, urging that the ticket include a woman.

“Democratic victory in 2020 will depend on record-breaking participation by women,” said the letter, which was also signed by dozens of additional female leaders. “Women are the backbone of the Democratic Party. Women are a majority of Democratic voters, volunteers and donors.”

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The letter also called on both candidates to stock their Cabinets and top administration positions with women, make it a priority to meet with women, and outline a “first 100 days” policy agenda that focuses on issues such as paid family leave and pay equity.

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Biden’s decision to make his announcement relatively early in the campaign, rather than simply picking a woman when the time came, also suggests he thought it was important to send a signal at this juncture. The aim may have been in part to set up a contrast with Trump, who has made various derogatory comments about women.

It also reflects a shift in Democratic politics, where notions of identity and representation have gained power in recent years.

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“Until recently, the idea of choosing somebody other than a white man was just not part of the calculus,” said Joel Goldstein, a law professor at Saint Louis University who wrote a book on the vice presidency. “Nobody has ever previously said, ‘I will pick a woman.’ ”

When Democratic presidential nominee Walter Mondale selected then-Rep. Geraldine Ferraro (N.Y.) to be his running mate in 1984, a burgeoning women’s movement pushed him to make the move, Goldstein said. “They were threatening floor fights if they didn’t pick a woman,” he said.

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But Mondale made the selection with his back against the wall. The economy was recovering, making his challenge to President Ronald Reagan more difficult, and his advisers wanted to “reshuffle the deck” with an unusual pick, Goldstein said.

And 24 years later, when Republican John McCain picked Sarah Palin, he did so after it became clear that his first choice, Sen. Joe Lieberman — a Democrat from Connecticut — would cause a brawl at the convention. At the time McCain trailed Democratic nominee Barack Obama, a historic candidate running an electrifying campaign.

Biden, in contrast, is leading in the polls, so his announcement is hardly a desperation move. And his age means his choice of running mate will get extra scrutiny.

“I think people will look at any pick that Joe Biden makes — because of his age and because of the moment — with an increased sense of responsibility,” Weingarten said. “One should always do that about a vice president.” But that’s especially true, she said, since Biden, Sanders and Trump “are all in their 70s.”