WASHINGTON - Democrats on Tuesday pledged to muscle past the political sniping and policy feuds that have ensnared President Joe Biden's economic agenda for months, raising fresh hopes that Congress and the White House could soon adopt trillions of dollars in new, long-promised federal spending.

The fresh glimmer of hope arrived after another day of private talks between Biden and his own party's warring liberal and moderate factions. From these and other gatherings, Democratic leaders emerged newly optimistic that they could reach a deal as soon as this week on an overhaul to federal health-care, education, climate and tax laws.

"There was universal, universal agreement in that room, that we have to come to an agreement, and we got to get it done and want to get it done this week," Senate Majority Leader Charles Schumer, D-N.Y., said after one of the meetings, a lengthy party lunch he later described as "spirited."

"The pace has picked up," Schumer said. "The desire to get it done is strong."

At the center of the stalemate is a battle over the proposal's price tag. Left-leaning Democrats, led by Sen. Bernie Sanders, I-Vt., and Rep. Pramila Jayapal, D-Wash., have sought as much as $3.5 trillion over 10 years for a sprawling package that expands Medicare benefits, invests new sums to combat global warming and provides paid leave and tax credits to workers and families. But two key centrists, Sens. Joe Manchin III, D-W.Va., and Kyrsten Sinema, D-Ariz., have objected to the overall cost and vast policy scope favored by their liberal counterparts.

The standoff at times has proven acrimonious, pitting lawmakers like Sanders and Manchin against each other in a public war of words. Manchin specifically has pushed for a package closer to $1.5 trillion, forcing left-leaning members to make what they regard as unacceptable concessions. Some in the party instead believe a $2 trillion proposal is more realistic, though they still must sell Manchin and Sinema on the idea. In a sign of how quickly things are moving, though, Manchin and Sanders met Tuesday for their second gathering in two days as they try to hammer out a deal.

No matter its final size, Democrats require unanimity to proceed, since an objection by even a single party lawmaker could prove fatal in the Senate, where they possess only a tiebreaking majority. Much work remains to address their myriad differences, but some Democrats still appeared to embrace a new optimism Tuesday that they are approaching the next phase in what has been a marathon debate.

"We're all talking. Everybody's talking," Manchin told reporters. Asked what he told lawmakers at Senate Democrats' weekly lunch, he replied: "We have to come to an agreement one way or another."

The flurry of activity foreshadows what could be a frenetic, final sprint to end the year. Time is running out if lawmakers hope to deliver on Biden's economic agenda before 2022, when the fight over fiscal issues could grow even more intense entering a midterm election year that will determine control of Congress in the final half of Biden's first term.

Sensing the electoral stakes, the president is set to hit the road Wednesday, visiting Scranton, Pa., his birthplace, to pitch voters again on his vision. Previewing the trip, White House press secretary Jen Psaki said Biden intends to "talk about growing up in Scranton and the way his experience there influenced his values and his belief that we need an economy that works for working people."

The president's has put his full political weight behind two separate but related packages. The first is a bipartisan, roughly $1.2 trillion plan to improve the nation's roads, bridges, pipes, ports and Internet connections, which cleared the Senate earlier this year. The second is a broader, still-forming tax-and-spending package that Democrats hope to adopt using a legislative tactic that allows them to sidestep a Republican filibuster.

Politically, it appears neither legislative effort can advance without the other. Liberals have held up the infrastructure bill to maximize negotiations over additional spending, but they cannot advance that second package without centrists' support. With only slim voting majorities, Democrats have no room for error in the face of sustained Republican opposition, which GOP lawmakers reiterated in clear terms Tuesday.

"Every day, American families are feeling real pain because of the reckless and inflationary policies that Democrats have already rammed through," Senate Minority Leader Mitch McConnell, R-Ky., said during a speech on the chamber floor.

For Democrats, though, the urgent question is how large that second package should be. In recent days, Biden has told Democrats that they have no choice but to compromise from their initial $3.5 trillion price tag - and potentially come to terms with a smaller, roughly $2 trillion proposal. The cuts mean party lawmakers must rework or abandon some of their core policy priorities, a delicate process that continued amid a flurry of behind-the-scenes talks Tuesday.

"We are having discussions about what level of funding will be available - that is, that we can pass, that we can get the votes for," House Majority Leader Steny H. Hoyer, D-Md., said at a news conference Tuesday. "We are working very hard to find out how much we have, and what can we do in the most efficient and effective way to accomplish the build back better agenda."

At the White House, Biden began his afternoon by meeting with lawmakers including Jayapal, the leader of the Congressional Progressive Caucus. Exiting the gathering, Jayapal said she felt "very good" that the "majority" of the programs backed by the left-leaning bloc would be in the final deal.

"The president is the inspirer, he is the closer, he is the convincer, the he is the mediator-in-chief. He really is doing a phenomenal job," she said.

Another attendee, Rep. Ro Khanna, D-Calif., said many of their initial proposals are still on the table. That includes an expansion to Medicare, for example, and a program that provides universal prekindergarten for children, the Democrat later said during an interview on CNN. But liberals may be forced to accept a number of cuts, he said, including some new income requirements on a plan to offer enhanced child tax credits to families.

At the Capitol, meanwhile, Senate Democrats emerged from their own private lunch Tuesday expressing a sense of optimism about their ability to secure a new framework this week. But it remained unclear exactly how detailed such a deal might be, and how close the full caucus - including Sinema, a centrist holdout who did not attend the session - is to backing it.

"Closer than you think," Sen. Thomas R. Carper, D-Del., said Tuesday about the broader Democratic caucus. "What do they say about looking in your rearview mirror. . . . The object may be closer than you think. This is definitely doable. I am encouraged."

But the process to slim down the bill - a proposal Democrats see as transformative and central to the contemporary political legacy - at times has proven personal and fraught. Tempers had flared over the past week, particularly between Manchin and Sanders, the leader of the Budget Committee and the chief architect of the $3.5 trillion plan.

Publicly and privately, Manchin has sought to reduce the package by more than half, while removing some of its critical components. In doing so, he has opposed one of its more signature elements to encourage power providers to produce cleaner energy. He has called on Democrats to scale back child tax credits that many Democrats believe should be more robust. And he has objected to some of the expansions to Medicare favored by Sanders and his allies, jeopardizing Democrats' ability to expand seniors' coverage to include new dental, hearing and vision benefits.

The potential cuts have only infuriated Sanders, who penned an op-ed in a West Virginia newspaper last week that pitched the benefits of the $3.5 trillion plan - and some of the components that Manchin opposes - directly to the moderate senator's own voters. Manchin responded in kind, issuing a rebuttal later Friday that reaffirmed his concerns about new federal spending at a time when "millions of jobs are open, supply chains are strained and unavoidable inflation taxes are draining workers' hard-earned wages."

"This isn't the first time an out-of-stater has tried to tell West Virginians what is best for them despite having no relationship to our state," Manchin said. "To be clear, again, Congress should proceed with caution on any additional spending, and I will not vote for a reckless expansion of government programs. No op-ed from a self-declared Independent socialist is going to change that."

But tempers seem to cool quickly and in a sign of how things appear to be moving quickly, the duo met Monday and Tuesday as Democrats began a new race for a deal.

The Washington Post's Mike DeBonis and Seung Min Kim contributed to this story.