WASHINGTON — President Biden huddled with key Democrats on Sunday to iron out crucial spending and tax provisions as they raced to wrap up their expansive social safety net legislation before his appearance at a U.N. climate summit next week.

Speaker Nancy Pelosi of California said Democrats were close to completing the bill, displaying confidence that the negotiations over issues like paid leave, tax increases and Medicare benefits that have bedeviled the party for months would soon end.

“We have 90 percent of the bill agreed to and written. We just have some of the last decisions to be made,” Ms. Pelosi said on CNN’s “State of the Union,” adding that she hoped to pass an infrastructure bill that had already cleared the Senate and have a deal in hand on the social policy bill by the end of the week. “We’re pretty much there now.”

Her comments came as Mr. Biden met with Senators Chuck Schumer of New York, the majority leader, and Joe Manchin III of West Virginia, one of the critical centrist holdouts on the budget bill. The White House called the breakfast at Mr. Biden’s Wilmington home a “productive discussion.”

For weeks, intraparty divisions over the scope and size of their marquee domestic policy plan have delayed an agreement on how to trim the initial $3.5 trillion blueprint Democrats passed this year. In order to bypass united Republican opposition and pass the final bill, Democrats are using an arcane budget process known as reconciliation, which shields fiscal legislation from a filibuster but would require every Senate Democrat to unite behind the plan in the evenly divided chamber. The party’s margins in the House are not much more forgiving.

Facing opposition over the $3.5 trillion price tag, White House and party leaders are coalescing around a cost of up to $2 trillion over 10 years. They have spent days negotiating primarily with Mr. Manchin and Senator Kyrsten Sinema, Democrat of Arizona and another centrist holdout.

House Democratic leaders hope to advance both a compromise reconciliation package and the $1 trillion bipartisan infrastructure package. Liberals have so far balked at voting on the bipartisan deal until the more expansive domestic policy package — which is expected to address climate change, public education and health care — is agreed upon.

But Democrats are facing a new sense of urgency to finish the legislation before Mr. Biden’s trip to a major United Nations climate change conference, where he hopes to point to the bill as proof that the United States is serious about leading the effort to fight global warming.

“The president looked us in the eye, and he said: ‘I need this before I go and represent the United States in Glasgow. American prestige is on the line,’” said Representative Ro Khanna, Democrat of California, who met with Mr. Biden last week at the White House.

Democrats are also increasingly eager to deliver the bipartisan legislation to Mr. Biden’s desk before elections for governor in Virginia and New Jersey on Nov. 2, to show voters the party is making good on its promise to deliver sweeping social change. And a number of transportation programs will lapse at the end of the month without congressional action on either a stopgap extension or passage of the infrastructure bill, leading to possible furloughs.

The legislation is expected to include a one-year extension of payments to most families with children, first approved as part of the $1.9 trillion pandemic relief plan, as well as an increase in funds for Pell grants, support for home and elder care, and billions of dollars for affordable housing. It would also provide tax incentives to encourage use of wind, solar and other clean energy.

While aides cautioned that details were in flux, the plan is also expected to address a cap on how much taxpayers can deduct in state and local taxes, a key priority for Mr. Schumer and other lawmakers who represent higher-income residents of high-tax states affected by the limit.

But negotiators on Sunday were still haggling over a number of outstanding pieces, including the details of a federal paid and medical leave program — already cut to four weeks from 12 weeks — Medicaid expansion and a push to expand Medicare benefits to include dental, vision and hearing. With Mr. Manchin pushing for a $1.5 trillion price tag, Democratic officials are urging for him to accept more spending in order to avoid dropping other programs.

Mr. Biden raised the prospect of an $800 voucher to help grant access to those benefits, but he said during a CNN town hall event on Thursday that he believed Mr. Manchin and Ms. Sinema both had reservations about the program. Talks are also continuing with moderate Democrats in both chambers over a proposal that would help lower the cost of prescription drugs.

While Ms. Sinema has refused to embrace increasing either the corporate or individual tax rates, she has privately committed to enough proposals that would fully fund up to $2 trillion in spending. The details of those specific proposals, however, remain unclear.

Ms. Sinema and Senator Elizabeth Warren, Democrat of Massachusetts, have spoken repeatedly about a number of alternatives that would ensure that corporations and wealthy people pay more in taxes without addressing those rates, according to an aide, some of which Ms. Warren had initially proposed earlier this year.

Ms. Pelosi said on Sunday that she expected Senate Democrats to unveil as soon as Monday a proposal that would raise hundreds of billions of dollars to pay for the bill with a wealth tax on American billionaires.

Liberals are also worried about the package’s ability to sufficiently address the toll of climate change, particularly after Mr. Biden publicly conceded that a $150 billion program to more quickly transition to clean energy would be altered because of fervent opposition from Mr. Manchin.

“Joe is open to my convincing him that I can use it to increase environmental progress without it being that particular deal,” Mr. Biden said during the CNN town hall.

“I know there will be disappointment as some of the threads in this social spending package get pulled out,” Representative Jared Huffman, Democrat of California, said in a recent interview, adding that he had pressed Mr. Biden repeatedly on ensuring the strength of the package’s climate provisions. But, he added, “it’s just different on the climate side of this package — we will not have a second or third chance.”