WASHINGTON — Democrats struggled on Wednesday to cobble together an expansive social policy, climate change and tax increase plan, reaching for an elusive compromise on President Biden’s agenda even as centrist holdouts in the Senate refused to embrace key components.

House Democrats held out hope of a breakthrough on the domestic policy measure that could also put a $1 trillion bipartisan infrastructure bill on track for a vote as soon as Thursday. But as they ran short on time, party members were haggling over which programs should be included and how to pay for the plan.

Senator Joe Manchin III of West Virginia objected to a leading proposal to finance the measure by levying huge taxes on the unrealized capital gains of billionaires, an idea that has also raised concerns with other Democrats who question its feasibility. His opposition also prompted negotiators to concede privately late Wednesday that a plan to establish a federal paid family and medical leave program would most likely be jettisoned, infuriating proponents who said they would fight to salvage the benefit.

And leading progressives demanded to see the legislative text of the bill before they would support a vote on the infrastructure measure, which has been stalled for months as Democrats have searched — so far in vain — for a compromise on their broader social policy plan.

“We’ve got a struggle on our hands,” said Senator Bernie Sanders, the Vermont independent who leads the Senate Budget Committee, after meeting with Mr. Biden at the White House.

The effort unfolded as Mr. Biden was set to leave Thursday morning for a trip to Europe and on to Scotland for a climate summit on Sunday, where he had hoped to point to a deal on his marquee legislation on as evidence of the United States’ commitment to tackling climate change. Democrats were also eager to see action on the infrastructure measure by next week, to bolster Terry McAuliffe in Virginia’s tight governor’s race on Tuesday.

Speaker Nancy Pelosi of California privately instructed top committee leaders to wrap up work on whatever could be agreed upon in the social policy measure and have legislative language ready for a meeting of the House Rules Committee on Thursday. Democratic leaders also scheduled a morning caucus meeting, as they sought to rally their frustrated and weary rank-and-file members behind the plan.

“An agreement is within arm’s length,” Senator Chuck Schumer of New York, the majority leader, said as he continued to prod his members toward what he called the “legislative sweet spot” that has eluded negotiators over weeks of feverish negotiations.

Because of united Republican opposition, Democrats are pushing the social policy plan through Congress using a special budget process known as reconciliation, which shields tax and spending legislation from a filibuster. But Democrats were having a hard time cobbling together a bare majority to pass the bill given their slim margins of control, which will require the backing of all 50 of their caucus members in the Senate and all but a few in the House.

Much of the attention focused on Mr. Manchin and Senator Kyrsten Sinema of Arizona, another centrist who has objected to key elements of the bill. The pair huddled for more than an hour on Wednesday on Capitol Hill with senior White House aides, and later could be seen on the Senate floor being buttonholed by a succession of colleagues lobbying them on various aspects of the package.

Lawmakers were petitioning Mr. Manchin in particular over his concerns with the leave program and the billionaires tax, which top Senate Democrats unveiled on Wednesday.

“I don’t like the connotation that we’re targeting different people,” Mr. Manchin said of the proposed tax.

Representative Richard E. Neal of Massachusetts, the top tax-writer in the House, warned reporters that there was “a lot of angst” over the proposal.

Mr. Manchin’s objections to the leave program could be a final blow for the plan, which had already been whittled down to four weeks from a dozen as Democrats trimmed the package.

Proponents vowed to keep pressing Mr. Manchin to accept some version of the program.

“We are not going to let one man tell millions of women in this country that they can’t have paid leave,” declared Senator Patty Murray of Washington, a member of Democratic leadership.

Senator Kirsten Gillibrand of New York, the provision’s biggest champion in the Senate, said Mr. Manchin had assured her that he was keeping an open mind.

But Mr. Manchin’s concerns could prove insurmountable.

“I’m looking at everything,” Mr. Manchin said Wednesday evening. “But to put this into a reconciliation bill — it’s a major policy — is not the place to do it.”

The grueling negotiations against a calendar crunch mirrored the situation Democrats faced just a month ago as they tried and failed to make progress on both major bills that are carrying Mr. Biden’s domestic agenda. In late September, House leaders planned a vote on the infrastructure bill in line with a promise to moderates who had demanded it, but encountered a blockade by progressives, who refused to vote for that measure without a vote on the social policy plan.

A frenzied week of negotiation yielded some narrowing of differences, but no breakthrough, and Democrats agreed to set a new deadline of Oct. 31 for action.

That date is now days away, and while Democrats are much closer to a compromise on the reconciliation package, it was clear on Wednesday that they had not made sufficient progress to guarantee a vote on either measure.

Without passage of the infrastructure bill by Sunday, lawmakers will have to take up a stopgap bill to avoid the expiration of federal transportation programs that are set to lapse.

Representative Pramila Jayapal of Washington, a leading progressive, said on Wednesday that liberals in the House must see the legislative text of the social policy bill before they would vote for the public works measure, saying the two should move in tandem.

“I just don’t want to see us make the mistake of not having the legislative text,” she added. “Let’s do them both together. Let’s get it right. Let’s make sure there are no misunderstandings, because there have been too many misunderstandings.”

Democrats are coalescing around a measure that would extend monthly tax credit payments to families with children, create a federally guaranteed prekindergarten program and expand financial assistance for child care, home health care and worker training and housing. But a number of provisions remain in flux, including a push to expand Medicare to cover dental, vision and hearing benefits, a priority for Mr. Sanders and other liberals.

As it stands, passage of the bill, which is also expected to include roughly $500 billion for climate and environmental provisions, could technically achieve Mr. Biden’s ambitious promise for lowering U.S. emissions, though many potential obstacles would remain, according to an analysis by Rhodium Group, an independent policy research firm. Those provisions would include about $300 billion in tax incentives to promote renewable energy and electric vehicles.

Lawmakers have also scaled back their ambitions for providing health coverage for poor adults whose states chose not to expand Medicaid, and are considering a more temporary fix. While negotiations continue, several people familiar with the discussions said the package may include three years of funding for poor residents of those states to get federal assistance buying health plans under the Affordable Care Act.

The plans would be effectively free, and would come with subsidies to lower deductibles and other forms of cost sharing. An estimated 2.1 million poor Americans are uninsured in those states, mostly in the South, and insuring them has been a key priority for lawmakers like Representative James E. Clyburn of South Carolina, the No. 3 House Democrat, and Senators Raphael Warnock and Jon Ossoff of Georgia. It has also been a priority for Ms. Pelosi, who views the measure as key to cementing the framework of the Affordable Care Act.

The compromise has disappointed some lawmakers and advocates who had hoped for a permanent coverage expansion — and for those who thought a federal Medicaid program would provide better safeguards for low-income patients than health exchange plans.

Reporting was contributed by Jonathan Weisman, Margot Sanger-Katz, Coral Davenport and Catie Edmondson.