Democrats lobby Manchin as they race to seal a deal on their domestic agenda.

Senator Kirsten Gillibrand, Democrat of New York, reached out to him asking for help extending federal paid leave.

The No. 3 House Democrat, Representative James E. Clyburn of South Carolina, and both Georgia senators, Raphael Warnock and Jon Ossoff, have made personal pleas for him to drop his opposition to a Medicaid expansion that would help their constituents.

As Democrats push for a compromise on their marquee domestic priority, Senator Joe Manchin III of West Virginia, the most outspoken holdout on the plan and a crucial swing vote, has become the object of intense lobbying by his colleagues.

In an evenly divided Senate, the fate of President Biden’s domestic agenda could hinge on any Democratic senator willing to withhold their vote. And while Mr. Manchin is not the only centrist who has raised concerns, his lengthy bill of particulars has made him a popular target for lawmakers trying to salvage provisions in the rapidly shrinking plan.

In recent days, Mr. Manchin has publicly objected to major components of the bill, including expansions of Medicare and Medicaid, a federal paid leave program, two major climate provisions, and a proposal to empower the Internal Revenue Service to obtain data for customers’ bank accounts as part of an effort to crack down on unpaid taxes and raise revenue to pay for the package.

Mr. Manchin, whose demand that the overall package not exceed $1.5 trillion has driven a frenzied effort to cut down the cost, has maintained that he is keeping an open mind out of fairness to Mr. Biden and Democratic leaders.

His maneuvering has prompted rumors that he might soon leave the Democratic Party, but he describes it as part of his political identity, no matter his affiliation.

“Do you think by having a D or an I or an R is going to change who I am?” Mr. Manchin said on Tuesday during an event hosted by the Economic Club of Washington. “I don’t think the Rs would be any more happier with me than Ds are right now.”

“I don’t know where in the hell I belong,” he concluded, drawing laughter from the audience.

These days, he can usually be found huddled with other Democrats who are seeking his support — or to change his mind — on a component of the social policy bill. On Monday evening, it was Senator Chuck Schumer of New York, the majority leader, along with Senators Thomas R. Carper of Delaware, Maria Cantwell of Washington and Ron Wyden of Oregon, all seeking to preserve a fee on methane emissions to which Mr. Manchin had objected.

Democrats have good reason to court Mr. Manchin’s support. They well remember how difficult it was to win his vote on the $1.9 trillion pandemic aid plan enacted earlier this year, and his last-minute effort to slash the size of the unemployment benefits included in it. (While Mr. Manchin ultimately voted yes after winning concessions, the grueling negotiating session led to the longest open Senate vote in modern history.)

Seven months later, he and Senator Kyrsten Sinema, Democrat of Arizona and another holdout, are similarly situated to cause problems for the bill if they cannot be won over.

“I’m talking to Senator Manchin and all my colleagues,” Mr. Warnock said. “They know that Medicaid expansion is a top priority for me.”