“The case that the White House is making is that compromise requires everybody giving a little. That’s the stage we’re in,” said White House press secretary

Jen Psaki

ahead of Mr. Biden’s meeting with the House Democratic caucus. “No matter where we end, if we can get something done here, we’re going to have a historic piece of legislation pass Congress.”

House Speaker

Nancy Pelosi

(D., Calif.) said that the House would take up the infrastructure bill on Friday, and Democrats held a caucus meeting in the morning. The infrastructure bill would fund improvements to roads, bridges, ports and expanded broadband Internet access, while the social policy and climate bill would fund a variety of education, healthcare, child-care and climate initiatives.

Top Democrats and White House officials held a marathon session of negotiations on Thursday to try to resolve the disagreements, with more meetings expected Friday. Central to the discussions are centrist Sens. Joe Manchin (D., W.Va.) and Kyrsten Sinema (D., Ariz.), who have each opposed the $3.5 trillion price tag for the social policy and climate bill.

Mr. Manchin said on Thursday that he could support spending $1.5 trillion on the social policy and climate bill, far below what progressive Democrats have pushed for. Some aides expect the new agreement to be worth roughly $2 trillion.

“We’re going to come to an agreement. I’m trying to make sure they understand I’m at 1.5 trillion. I think 1.5 trillion does exactly the necessary things we need to do,” Mr. Manchin told reporters after a meeting with White House aides late Thursday night.

The tortured effort to unite the party around the two pieces of legislation reflects the difficulties of wielding the very narrow majorities Democrats hold in Congress. Democrats cannot afford a single defection in the 50-50 Senate, and they can lose no more than three votes in the House, meaning leadership must build nearly unanimous consensus for the policies in the party. They are pursuing a process called reconciliation to approve the social policy and climate bill without GOP support in the Senate, where legislation would otherwise require 60 votes to advance.

Democrats are hoping to secure a broad agreement with centrists on the social policy and climate bill to reassure progressives about that effort’s fate.

Rep. Pramila Jayapal

(D., Wash.), the chairwoman of the Congressional Progressive Caucus, said that progressives could support the infrastructure bill if there was a firm commitment on the other legislation. She has previously said she wanted to see the social policy and climate bill passed before the infrastructure bill.

“The reason I want a vote is I want to be assured that there’s no delay, and that there’s no misunderstandings about what we agreed to, and so if there’s something else that’s short of a vote that somebody can offer me that gives me those same assurances, I want to listen to that,” she said.

As part of an earlier agreement with House centrists, Mrs. Pelosi had agreed to hold a vote on the infrastructure bill this past Monday, which she later pushed to Thursday. Delaying the vote again frustrated centrist House Democrats, who have pushed to pass the infrastructure bill, while negotiations continue on the social policy package.

“When Iowans tell me they are sick of Washington games, this is what they mean,”

Rep. Cindy Axne

(D., Iowa) said Thursday night. “Some in my party are insisting that we wait to put shovels in the ground and pass the largest investment in rural broadband in U.S. history until every piece of our agenda is ready.”

The failure to pass the infrastructure bill before midnight led to a lapse in authorization for the nation’s transportation programs, putting roughly 3,700 federal employees temporarily on furlough, according to a Department of Transportation spokeswoman. Lawmakers have discussed a short-term patch to continue to reauthorize the transportation programs, while negotiations continue on the broader infrastructure bill.

Democrats hope to tackle an array of policy areas in the healthcare, education and climate package, drafting a bill in the House that would offer universal prekindergarten, two years of free community college, expand Medicare and implement many other party priorities. Meeting Mr. Manchin’s desire for a $1.5 trillion bill would require paring back or eliminating many of the provisions.

Some liberal Democrats were unfazed by Mr. Manchin’s demand. Sen. Elizabeth Warren (D., Mass.) said that negotiations among Democrats would focus on the specific initiatives that they could unify around.

“A lot of different people have given a lot of different numbers over the past several months. But what we’ve all talked about as Democrats is things we need to get done and we need to do with a realistic price tag,” she said.

Though 19 Senate Republicans backed the infrastructure bill and at least some House Republicans are also expected to do so, it isn’t clear whether there will be enough GOP support to offset opposition from liberal Democrats.

—Sabrina Siddiqui contributed to this article.

Write to Andrew Duehren at andrew.duehren@wsj.com , Kristina Peterson at kristina.peterson@wsj.com and Eliza Collins at eliza.collins+1@wsj.com.