WASHINGTON—House Democrats released an updated version of the party’s social spending and climate package, adding back a paid-leave program that had previously fallen out of the bill and including a measure sharply raising the $10,000 cap on the state and local tax deduction.

The House bill, which top Democrats want to bring up to a vote in the chamber soon, is the latest proposal in the monthslong negotiations among Democrats over President Biden’s agenda. But it is set to face changes in the Senate, where Sen. Joe Manchin...

WASHINGTON—House Democrats released an updated version of the party’s social spending and climate package, adding back a paid-leave program that had previously fallen out of the bill and including a measure sharply raising the $10,000 cap on the state and local tax deduction.

The House bill, which top Democrats want to bring up to a vote in the chamber soon, is the latest proposal in the monthslong negotiations among Democrats over President Biden’s agenda. But it is set to face changes in the Senate, where Sen. Joe Manchin (D., W.Va.) has objected to the inclusion of a paid-leave benefit.

The bill includes a variety of measures, proposing a universal prekindergarten program for three- and four-year-olds, subsidies for child-care and healthcare costs, and tax credits for reducing carbon emissions, among other measures. The House text leaves many of those items, as well as proposed corporate minimum taxes and surtaxes on high-income individuals, largely unchanged from the White House’s $1.85 trillion framework released last week.

In recent days, Democrats have rushed to resolve a final set of issues on prescription drug pricing, immigration and the capped deduction on state and local taxes.

Democrats’ loss in Tuesday’s gubernatorial race in Virginia has added fresh urgency for the party to close in on an agreement. Lawmakers hope that passing the broad legislation in the House will also allow them to move forward with a separate, roughly $1 trillion infrastructure bill that progressives have held up for months as leverage.

“People want us to get things done, and that’s what I’m continuing to push very hard for the Democratic Party to move along and pass my infrastructure bill and my Build Back Better bill,” President Biden told reporters at the White House on Wednesday.

Democratic lawmakers left a closed-door meeting on Tuesday saying they expected to vote on both bills by the end of the week, though some centrist Democrats have demanded time to review and analyze the education, healthcare and climate legislation.

The House plan would raise the $10,000 cap on the state and local deduction to $72,500, starting in tax year 2021. It would also extend that higher cap through 2031, beyond its scheduled expiration after 2025. That feature means that the higher taxes in future years would pay for the short-term tax cut, at least in congressional budgeting terms.

Sen. Joe Manchin (D., W. Va,) voiced opposition to a paid-leave provision being added to the party’s education, healthcare and climate bill.

Photo: J. Scott Applewhite/Associated Press

Democrats representing high-tax states such as New York and New Jersey have pressed for a change to the $10,000 cap, which Republicans put in place as part of their 2017 tax law. But other Democrats have criticized efforts to repeal or lift the cap, arguing that it will primarily benefit the wealthy.

House Democrats also added new protections for immigrants in the country illegally to the bill, proposing to shield the immigrants from deportation for five years and provide a five-year, renewable work authorization. Some Hispanic lawmakers are seeking to include a pathway to citizenship in the legislation. Previous immigration proposals have run into difficulty with the Senate’s procedural rules.

The paid-leave plan, which will cost roughly $200 billion, calls for four weeks of parental, sick or caregiving leave, shorter than the 12 weeks Democrats had initially hoped to include in the package. Under the program, the government would provide Americans a share of their wages while they take leave.

The U.S. is one of few industrialized nations without a national paid leave program for new parents. According to the Bureau of Labor Statistics, as of March 2021, 23% of civilian workers in the U.S. had access to paid family leave and 89% had access to unpaid family leave.

The measure faces an uphill battle in the Senate, where Mr. Manchin said Wednesday he continued to oppose its inclusion in the bill. Democratic leaders had stripped it out of the bill because of the West Virginia Democrat’s continued opposition, which other Democrats, several of them women, have pushed him to reconsider.

In a meeting with House Democrats, Speaker Nancy Pelosi (D., Calif.) stressed the importance of paid leave, and said she wanted to include it in the legislation, according to two people who heard the remarks. House Ways and Means Chairman Richard Neal (D., Mass.) said that he and Mrs. Pelosi decided to update the House bill with four weeks of paid leave in an early morning phone call on Wednesday, despite not having a commitment from all 50 senators that they would support it.

Mr. Manchin said Wednesday he “had no idea” Mrs. Pelosi planned to put the measure back in the House legislation, which is being passed through a process called reconciliation so that Democrats can skirt Republican opposition to it in the Senate. He said he wanted to work with Republicans on paid leave in separate legislation.

“It should not be in a reconciliation” bill, he said.

Democrats have dropped several measures—including free community college, a program aimed at pushing utilities to generate more clean energy and a series of top marginal-tax-rate increases—because of opposition from Mr. Manchin and other centrists.

President Biden discussed a $1.85 trillion social-spending and climate framework, calling the legislation "historic." Progressives have said they would block the infrastructure bill without a deal on the social-policy and climate bill. The Wall Street Journal Interactive Edition

Mrs. Pelosi has said she hopes to pass legislation in the House that will win the support of all 50 members of the Senate Democratic caucus. But the late-stage additions will likely complicate that effort.

Lifting the cap on the state and local tax deduction to $72,500 also faced quick criticism in the Senate. Sen. Bernie Sanders (I., Vt.) has sought to eliminate the cap permanently for households making up to about $400,000. Mr. Sanders called the new House plan regressive, and Sen. Bob Menendez (D., N.J.) said Wednesday that he and Mr. Sanders had reached an agreement in principle on a new compromise.

Sen. Michael Bennet (D., Colo.) said Democrats should alter the plan in the Senate so it provides less of a tax benefit to high-income Americans.

The House plan would cut taxes for more than 70% of households making more than $200,000, according to the Tax Foundation. Measured as a share of after-tax income, the largest benefits would go to households making between $500,000 and $1 million.

Because the break would start in 2021, the House plan would provide an immediate windfall to people filing their tax returns early next year. Many people in this income group have been assuming the $10,000 cap will be in place when setting their paycheck withholding and estimated taxes, which means they will get a larger-than-expected refund in early 2022.

The latest version of the House bill also adds a variety of tax provisions, many of which came out of the Ways and Means Committee but weren’t included in the prior version of the legislation or specifically mentioned in the White House framework.

For instance, the bill would create a new tax credit for companies that employ local news journalists, expand a tax break for employers that pay for child care and allow deductions for union dues and employee uniforms.

It would also limit some wealthy people’s ability to make contributions to tax-advantaged retirement accounts, once their account balance reaches $10 million, and add other restrictions on large retirement accounts.

Democrats restricted a proposed tax credit for plug-in electric vehicles. Earlier versions of the legislation had allowed the full credit for married couples making up to $800,000; that is now reduced to $500,000. The bill also no longer includes a provision that would lower a tax on private colleges’ endowment income.

—Andrew Restuccia contributed to this article.

Write to Andrew Duehren at andrew.duehren@wsj.com , Natalie Andrews at Natalie.Andrews@wsj.com and Richard Rubin at richard.rubin@wsj.com