The Biden administration is moving to close gaps between House and Senate Democrats on its roughly $2 trillion education, healthcare and climate package, so it can finalize the bill that serves as the centerpiece of the president’s agenda.

“We will work with every member of the Senate on this bill,” said Brian Deese, director of the National Economic Council, which advises the president on U.S. and global economic policy. “But I think that because of that work over several months, we really do now have a good understanding...

The Biden administration is moving to close gaps between House and Senate Democrats on its roughly $2 trillion education, healthcare and climate package, so it can finalize the bill that serves as the centerpiece of the president’s agenda.

“We will work with every member of the Senate on this bill,” said Brian Deese, director of the National Economic Council, which advises the president on U.S. and global economic policy. “But I think that because of that work over several months, we really do now have a good understanding of where the consensus lies.”

Mr. Deese, who was speaking on “Fox News Sunday,” didn’t offer a timetable for the bill to pass the Senate.

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To ensure revenue covers spending, Mr. Deese said the Biden administration is relying on “experts that have studied the provisions in this bill the most closely, the Congressional Budget Office as well as the Treasury Department.”

The CBO found that the bill would contribute $367 billion to the deficit over 10 years; Democrats have argued that revenue not captured in the CBO score shows the bill is more than fully paid.

On Friday, the House passed what the Biden administration has dubbed the Build Back Better bill by largely holding together the Democrats’ centrist and progressive wings. The package faces a more narrow path through the Senate, where lawmakers are evenly divided between Republicans and Democrats. Democrats are also planning to change or pare back some of the bill’s provisions in the coming weeks.

Sen. Jon Tester, a Montana Democrat, said he expected the legislation to ultimately pass with some changes. “It is going to come over from the House. There’s going to be some changes. I’m going to compare it to what Montana needs and that’s going to be where I focus on,” he said speaking on NBC’s “Meet the Press” on Sunday. “But look, we’re dealing with reasonable people here. I think we come up with a bill that is a very, very good bill, that works for states like Montana and other states.”

Mr. Tester singled out the bill’s provisions on child care, affordable housing, climate change and reducing healthcare and prescription drug costs as ones he supported. “I think people need to be open to compromise,” he said, pointing to the $1 trillion infrastructure bill that Mr. Biden signed into law earlier this month, with the support of some Republicans, as a model.

Republicans on Sunday repeated their criticisms of the legislation as pumping trillions of dollars into an economy already struggling with inflation. “This is an immense amount of money,” said New Hampshire Gov. Chris Sununu, speaking on CNN’s “State of the Union.” “Washington doesn’t seem to understand the concept of a budget.”

The house passage of the social spending and climate package has already inflamed an intraparty debate among Democrats about whether the bill gives overly generous tax benefits to high-income Americans.

At the center of the dispute is the House plan to raise the $10,000 cap on the deduction for state and local taxes to $80,000 through 2030. A small but committed group of lawmakers from high-tax states like New York and New Jersey have for years insisted on repealing the $10,000 cap, which Republicans put into place as part of the 2017 tax law.

But lifting the cap has tilted the tax benefits in the bill toward high-income Americans, alarming other Democrats who have themselves promised to raise taxes on the wealthy, not cut them. Republicans have also pilloried the provision, branding it as a handout to wealthy Democratic supporters.

Defenders of the House provision say it would provide needed tax relief to households in high-cost areas that they say Republicans politically targeted in the 2017 law. Some Democrats also fear that the cap would limit how high states could raise their own tax rates to pay for public services, though New York and New Jersey have raised rates since the cap went into law.

The state and local tax deduction is one of several issues in the bill that Democrats are hoping to iron out in coming weeks in the Senate, where the party will need unanimous support to approve the legislation. Sen. Joe Manchin (D., W.Va.) has demanded that Democrats cut four weeks of paid leave provided in the House bill, for example, and temporary immigration protections in the House bill may run afoul of Senate parliamentary rules.

Sen. Joe Manchin (D., W.Va.) has demanded that Democrats cut four weeks of paid leave provided in the House bill.

Photo: chris kleponis/Press pool

Sen. Kirsten Gillibrand (D, N.Y.) said on CBS’s “Face the Nation” that she was optimistic that the Senate bill would include paid family and medical leave, despite opposition from Mr. Manchin, a key Democrat holdout.

While the House provision includes four weeks of paid family and medical leave, the Senate bill includes none. Sen. Gillibrand said during conversations over the past three weeks, Sen. Manchin “has come a long way on paid leave” and that she was hopeful to make progress in the weeks ahead.

Any leave in this bill would likely be modest and a “Democrat-only proposal,” Ms. Gillibrand said, and would lay the groundwork for expanded leave in future legislation. “I’d be grateful if we could meet the House proposal. I think that would be a great start.”

The Biden administration is also facing a potential funding crunch for its current programs.

In September, President Biden signed a bill extending government funding through Dec. 3, averting a partial shutdown. The bill passed both the House and Senate with bipartisan support.

Earlier this month, Treasury Secretary Janet Yellen said the U.S. government could run out of resources to meet the nation’s obligations as soon as Dec. 15. The warning revived questions about how Congress will resolve a standoff about raising or suspending the federal borrowing limit.

—Andrew Duehren and Richard Rubin contributed to this article.

Write to Aruna Viswanatha at Aruna.Viswanatha@wsj.com and Nancy A. Youssef at nancy.youssef@wsj.com