WASHINGTON — President Biden and Democratic congressional leaders raced on Monday to strike a compromise on a domestic policy and climate package, pushing for a vote within days even as critical disagreements remained over health benefits, paid leave, environmental provisions and how to pay for the sprawling plan.

Negotiators were closing in on an agreement that could spend around $1.75 trillion over 10 years, half the size of the blueprint Democrats approved earlier this year, as they haggled with centrist holdouts in their party who are pressing to curtail the size of the bill.

They have coalesced around a plan that would extend monthly payments to families with children, establish generous tax incentives for clean energy use and provide federal support for child care, elder care and universal prekindergarten. An array of tax increases, including a new wealth tax for the country’s billionaires, would pay for the initiatives.

But a final deal remained elusive amid disputes over the details of potential Medicare and Medicaid expansions, a new paid family and medical leave program, programs to combat climate change and a proposal to lower the cost of prescription drugs. Top Democrats were also toiling to nudge the price tag up to $2 trillion, still far below the $3.5 trillion level they laid out in their budget plan.

Introducing a fresh wrinkle into the talks, Senator Joe Manchin III of West Virginia, a centrist who has led the effort to scale back the bill, was pushing to remove or modify a provision that would impose a fee on emissions of methane, a powerful planet-warming pollutant that leaks from oil and gas wells, according to two people familiar with the negotiations.

Mr. Manchin has already effectively killed the most powerful climate change provision in the package, a proposed $150 billion program that would replace coal- and gas-fired power plants with wind and solar power, though that money may be repurposed.

With Republicans uniformly opposed to the bill, Democrats cannot afford to lose even a single vote from their party in the 50-to-50 Senate, giving any senator the power to sink the plan over even a single provision. That has complicated the task facing Mr. Biden and party leaders as they prepare to muscle the bill through Congress using a special budget process known as reconciliation that shields it from a filibuster.

Administration officials and Democratic leaders hoped to have a deal secured before Mr. Biden appears at a United Nations climate conference in Glasgow that begins on Sunday, when he is expected to push for a stronger international response to counter global warming. Mr. Biden, who traveled to New Jersey on Monday to promote the legislation, told reporters that it would be “very, very positive to get it done before the trip.”

“It changes the lives of the American people,” Mr. Biden said during an appearance at a transit maintenance facility in Kearny, N.J., where he promoted the payments to families with children and child care assistance in the plan. “So let’s get this done — let’s move.”

The president also called for quick passage of a stalled $1 trillion bipartisan infrastructure bill, whose passage in the House could hinge on a deal on the reconciliation package. Liberals have refused to support the infrastructure measure until the broader policy package emerges.

Democrats want to pass the public works legislation — approved by the Senate over the summer — by Sunday, when a series of transportation programs is slated to expire. Its enactment would also hand the party a popular legislative achievement days before crucial governors’ elections in Virginia and New Jersey on Nov. 2.

But the reconciliation bill, widely seen as the last significant piece of legislation with a chance of passing this year, is a heavy lift.

“No one ever said passing transformational legislation like this would be easy, but we are on track to get this done,” said Senator Chuck Schumer of New York, the majority leader.

Mr. Manchin, who huddled with Mr. Biden and Mr. Schumer at the president’s home in Wilmington, Del., on Sunday, remained a driving force in the talks as he pressed his objections to key components of the plan. He was opposed to the paid leave program and to proposals to expand Medicare and Medicaid, according to officials briefed on the negotiations.

“There’s a lot of concerns. We have a lot of different things,” Mr. Manchin told reporters on Monday, saying he was still determined to keep the cost of the package to no more than $1.5 trillion. “We’re just looking at how we put it together.”

Mr. Manchin, a defender of his state’s coal industry, is also the most prominent opponent to key climate provisions that have long been a priority for the majority of his caucus. His sign-off on a compromise was seen as vital as Mr. Biden prepared to appear on the world stage to promote the U.S. commitment to tackling climate change.

A spokeswoman for Mr. Manchin did not respond to a request for comment on his position on the climate provisions. People briefed on the talks cautioned that details were still in flux.

Mr. Manchin is among the lawmakers who have expressed concerns with a push, led by Senator Bernie Sanders, the Vermont independent who is chairman of the Budget Committee, to expand Medicare to include dental, vision and hearing benefits.

“If we’re not being fiscally responsible, that’s really concerning,” Mr. Manchin said on Monday, stressing that he did not want to expand the federal health program for older people without addressing its financial stability.

He also expressed concerns about a push to cover a Medicaid expansion for the dozen states whose leaders have refused to expand the program under the Affordable Care Act. Its inclusion in the plan has been a priority for Democrats who represent those mostly southern states, such as Senator Raphael Warnock, Democrat of Georgia.

But West Virginia is among the states that expanded Medicaid and pay 10 percent of the cost. Mr. Manchin said it would be unfair for the federal government to cover the entire cost for states that had not done so, in essence rewarding them for holding out.

Democrats must also resolve their differences over how to finance the plan, after promising moderates in their ranks that it would be fully paid for. Moderates have balked at a proposal to lower the cost of prescription medicines by allowing Medicare to negotiate with drug companies on prices, prompting negotiators to look at narrower versions of that approach.

The details of the billionaires’ tax, an entirely new approach to taxing wealth, were also still being hammered out, although top Democrats said it could be released in the next couple of days. And negotiators were still working out how to meet a demand, primarily from New York and New Jersey lawmakers, to increase the amount of state and local taxes that people can deduct from their federal tax bill, which would benefit those living in states with high taxes.

The proposal, which could cost hundreds of billions of dollars, is a priority for Mr. Schumer, as well as moderate Democrats whose votes are needed to push through the reconciliation bill, including Representative Josh Gottheimer of New Jersey and Tom Suozzi of New York.

Because Senator Kyrsten Sinema of Arizona, another centrist holdout on the plan, has rejected increases to the corporate and individual tax rates, negotiators were discussing an array of alternatives, including the wealth tax, a global corporate minimum tax, a tax on what corporations report to shareholders and strengthening the I.R.S.’s ability to collect unpaid taxes.

“I think we will find a way that we can come together and have all 50 Democrats on board,” said Senator Elizabeth Warren of Massachusetts, who is heavily involved in the finance discussion. “We’re all trying to head in the same direction.”

Republicans, who adamantly oppose the domestic policy plan, railed against the new tax proposals under discussion on Monday.

“Our Democratic colleagues have become so tax-hike happy that they’re throwing spaghetti at the wall to see what sticks,” declared Senator Mitch McConnell of Kentucky, the minority leader. “Now they’re talking about rewiring the entire economy after a couple of days’ discussions on the back of an envelope. It’s a massive and untested change.”

Mr. Biden did not address the issue, or the state of negotiations, directly in his speech. But he outlined the benefits of his plans, leaning heavily into efforts to repair bridges and invest in passenger rail service, in an effort to win over voters in a state where Gov. Phil Murphy, a Democrat, is seeking re-election on Nov. 2.

Mr. Biden stopped earlier in the day at an early childhood education center, as part of an effort to spotlight the plan to provide federally guaranteed prekindergarten for 3- and 4-year-olds.

Karine Jean-Pierre, a White House spokeswoman, told reporters on Air Force One on Monday that Democrats had reached the point where they were negotiating “the nitty-gritty of the details” in the package.

“We have a strong base of agreement,” Ms. Jean-Pierre said. “We’re continuing to negotiate. We’re almost there.”

Coral Davenport, Jim Tankersley and Margot Sanger-Katz contributed reporting.