The deadline for House Speaker Nancy Pelosi to introduce Democrats’ healthcare, education, and climate package has been pushed back to Friday due to snags on immigration and the state and local tax deduction.

According to people familiar with her remarks, the California Democrat told House Democrats in a closed-door meeting on Thursday that she hoped the House would vote on the bill later that day, and then vote on the Senate-passed infrastructure bill on Friday. Progressives have been holding up the roughly $1 trillion infrastructure bill until they are satisfied with the language in the social-spending and climate-change legislation.

House Majority Leader Steny Hoyer (D-MD) announced late Thursday that the House would instead take up both pieces of legislation as early as Friday, thwarting Democrats’ efforts to move much of President Biden’s agenda through Congress. While the infrastructure bill would be sent to Mr. Biden for signature after passage in the House, the nearly $2 trillion healthcare, education, and climate bill will almost certainly be amended in the Senate.

A trio of House Democrats had threatened to vote against the bill unless it included more expansive immigration measures. After a meeting with Mrs. Pelosi on Thursday night, the group stated that they would continue to work on the issue, but not by changing the current House bill.

Previous immigration proposals for the bill have been thwarted by Senate procedural rules. Because Republicans are united in their opposition to the legislation, Democrats are using a procedure known as reconciliation to avoid the 60-vote threshold for most bills in the Senate. Reconciliation, on the other hand, constrains lawmakers by requiring them to pass measures directly related to the budget.

While negotiations on some measures will continue, the party has largely united around many of the bill’s core components. A universal prekindergarten program for 3- and 4-year-olds, child care and healthcare subsidies, and a series of carbon-cutting tax credits are all likely to pass through Congress. Tax increases on corporations’ foreign profits and high-income households are also on the table.

Democrats had previously proposed raising the $10,000 cap on the deduction for state and local taxes to $72,500 in a previous version of the House bill. Negotiations on the topic, however, continued throughout Thursday, with lawmakers closing in on an agreement that would raise the cap to $80,000 but allow the $10,000 cap to revert after nine years, according to the people.

According to a report released on Thursday by the Joint Committee on Taxation, the tax increases in the bill would raise slightly more than $1.5 trillion over a decade. The estimate, which took into account the effects of provisions such as a new corporate minimum tax and higher taxes on people earning more than $10 million, does not take into account other elements of the Democrats’ plan that would generate revenue. These include additional revenue from increased IRS tax enforcement and federal savings from changes in Medicare prescription-drug policy.

According to a White House official, the increased IRS enforcement could generate $400 billion in revenue, while the drug-pricing measures could save the government $250 billion. Other revenue estimates for the IRS enforcement plan, on the other hand, have found it to bring in less money.

Because of the combination of policies, Democrats are likely to have more than enough money to fund the original $1.85 trillion plan. House Democrats are raising the bill’s cost by reintroducing a roughly $200 billion paid-leave benefit for parents, the sick, and caregivers that had previously been dropped from the bill and may be dropped again in the Senate.

According to the Joint Committee on Taxation’s analysis, after accounting for tax breaks such as the expanded child tax credit, the tax portion of the bill would amount to a $945 billion tax increase over a decade. This represents a 1.8 percent increase in projected federal revenue, but it also represents a shift in who pays, with corporations and many high-income individuals paying more and many lower-income families paying less, particularly in the short run.