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Both houses of Congress approved the amendment in 1972, but it was not ratified by the states in time to be added to the Constitution. Ms. Gillibrand has been pushing a legal theory that the deadline for ratification is irrelevant and unconstitutional. All that remains, she argues, is for Mr. Biden to direct the national archivist, who is responsible for the certification and publication of constitutional amendments, to publish the E.R.A. as the 28th Amendment.

The move would almost certainly invite a legal challenge that would land in the Supreme Court. But Ms. Gillibrand wants Mr. Biden to use his presidential power while he still has it to force the issue, effectively daring Republicans to wage a legal battle to take away equal rights for women.

Ms. Gillibrand’s is just one of the many entreaties Democrats in Congress are making to Mr. Biden in the waning hours of his term to take bold action before President-elect Donald J. Trump and an all-Republican Congress take over in January. Some are pressing him to protect undocumented immigrants brought to the United States as children. Others are pushing for more clemency grants for incarcerated people.

Ms. Gillibrand has pleaded her E.R.A. case at every available opportunity. The third-term New Yorker has met with Jeffrey D. Zients, the White House chief of staff, and Anita Dunn, a former top adviser to Mr. Biden. Her request is for a five-minute meeting with Mr. Biden himself. She has used passing 30-second interactions in photo lines to personally pitch the president, so far to no avail.

Ms. Gillibrand has presented White House officials with fat binders full of legal research and polling, which have on the cover a printout of Mr. Biden posing as if he is on a Taylor Swift Eras Tour poster. (E.R.A. — get it?)

Ms. Gillibrand sat down with Minyon Moore, one of Vice President Kamala Harris’s top advisers and confidantes, to persuade Ms. Harris to champion the E.R.A. and asked her to speak to other White House officials about it. Still, nothing happened.

Undeterred, Ms. Gillibrand has continued to text and harangue and flatter, all in service of procuring a brief meeting with Mr. Biden to make a more comprehensive pitch.

“I’ve never done more legal analysis and work since I was a lawyer,” Ms. Gillibrand said. So far, she has been strung along.

“It’s ‘I’ll get back to you; I’ll get back to you.’ Everyone always says, ‘We love your arguments.’ I never know what the ‘but’ is.”

Kelly Scully, a White House spokeswoman, said senior administration officials had been discussing the proposal with lawmakers and other stakeholders.

“President Biden has been clear that he wants to see the Equal Rights Amendment definitively enshrined in the Constitution,” Ms. Scully said in a statement. “It is long past time that we recognize the clear will of the American people.”

The issue of the E.R.A., which was first proposed by women’s suffragists in 1923, is politically simple and legally complicated.

The Constitution states that proposed amendments must be passed by two-thirds majorities in both houses of Congress and ratified by three-quarters of the states. It makes no mention of a deadline for ratification, but Congress in modern times has typically included a seven-year clock for the states to sign on.

The House and the Senate approved the Equal Rights Amendment in 1972 with a seven-year deadline, and then when it had yet to be ratified by enough states by 1979, extended the deadline to 1982. By then, only 35 states had ratified it — still short of the three-quarters requirement. Since then, three more states — Nevada, Illinois and Virginia — have ratified the amendment, surpassing the threshold. But some other states have since rescinded their ratifications.

Conservative opponents have argued that the deadline was binding and the amendment is effectively dead. The Trump administration’s Justice Department instructed the archivist to take no action on the E.R.A. when Virginia in 2020 became the 38th and final state to ratify it.

The timeline issue has left the amendment in limbo. Many legal scholars, as well as attorneys general in 23 states, have argued that since there is no reference to a ratification deadline in the Constitution and no precedent for any state successfully reversing its approval, neither issue should stand in the way of the E.R.A. becoming the 28th Amendment and the law of the land.

The amendment has taken on greater urgency for many Democrats since the Supreme Court’s 2022 ruling in Dobbs v. Jackson Women’s Health Organization overturned abortion rights. Ms. Gillibrand and other abortion rights advocates argue that Mr. Biden has one last opportunity to enshrine reproductive rights into law for millions of women, and that there are strong constitutional and political arguments for him to do so.

Ms. Gillibrand said she was not dissuaded by the possibility that a conservative-leaning Supreme Court might strike down the amendment.

“This will ultimately be decided by the courts, but this is a moment in time where we should stand up for what we’re for,” she said. “If you’re waiting for the people you don’t like on the court to die, we’re all going to be dead. That’s too long to wait.”

Republicans have generally called the measure gratuitous, arguing that equal protections for women are included in the 14th Amendment. But they, too, have conceded that adopting the amendment could provide a new legal basis for protecting abortion after the overturning of Roe v. Wade, and said it should be opposed as a way to protect the lives of the unborn.

On Monday night, at the lightly attended White House Christmas party for members of Congress, Ms. Gillibrand said she tried again to press the issue when it was her turn to take a holiday photo with the Bidens.

“We know all about it,” Jill Biden, the first lady, assured her, expressing support for the idea. When Ms. Gillibrand approached Mr. Biden last summer on a photo line, he responded, “So you want me to make a big deal out of it?”

Ms. Gillibrand has even recruited former Secretary of State Hillary Clinton to the cause, and Mrs. Clinton has also spoken to the archivist, she said.

As they lobby for action, Ms. Gillibrand and other lawmakers are also trying to appeal to Mr. Biden’s ego, contending that amending the Constitution to explicitly bar sex discrimination could burnish his damaged legacy.

“I’ve been working really hard just on the audience of one, trying to make the case to him personally, to say, ‘I want this to be part of your legacy as a president I respect and admire; you should make this part of your legacy,’” she said. “I’m trying to give them all the legal reasons and political reasons.”

A new survey by Data for Progress, a progressive think tank, showed strong support for Mr. Biden taking action and that doing so would “cause a significant, positive shift among voters in terms of how they think Americans will look back on Biden’s presidency.” About 61 percent of likely voters, including about 87 percent of Democrats, said they supported Mr. Biden taking action to adopt the Equal Rights Amendment.

The poll also showed a 30-point increase in the percentage of voters who said they would look back on Mr. Biden’s presidency more favorably if he were to take action on the amendment.

Kate Kelly, the senior director of the women’s initiative at the Center for American Progress, who has been working on the issue for a decade, said she hoped that Mr. Biden and Democrats would not argue themselves out of taking action.

“This is the closest we’ve ever come for it being considered,” Ms. Kelly said. “I always tell people, ‘Think of what the other side would do if they were one signature away from changing the Constitution.’ We need that energy.”

Ms. Gillibrand is hopeful, even as she grows more frantic with less time left to get her meeting. Last month, 45 senators, including Chuck Schumer of New York, the majority leader, signed a letter to Mr. Biden calling on him to take action on the E.R.A.

“Inaction is action,” they wrote. “We must answer the call to uphold equality and strengthen women’s rights by certifying the E.R.A. We urge you to direct the archivist to publish the E.R.A. as the 28th Amendment without further delay.”

A group of about 100 House Democrats is also expected to send a similar letter to Mr. Biden, urging him to take action in the final days of his presidency.

“To President Biden: Take action now,” Representative Cori Bush, Democrat of Missouri, said at a rally last week outside the Capitol. “You have the opportunity to make equal rights a defining part of your legacy. All it takes is your signature.”