House Republicans on Wednesday pushed forward on their sweeping domestic policy bill, slogging through marathon drafting sessions that began Tuesday and stretched into the night as they haggled over Medicaid and tax cuts.

The meetings in three key committees, a crucial part of advancing what President Trump has labeled the “one big beautiful bill” carrying his agenda, came as Republican leaders raced to push the legislation through the House before a Memorial Day recess that begins at the end of next week.

Republicans are seeking to extend Mr. Trump’s 2017 tax cut and temporarily enact his campaign pledges not to tax tips or overtime pay. They want to partly offset the roughly $3.8 trillion cost of those tax measures — as well as plans to increase spending on the military and immigration enforcement — by making cuts to Medicaid, food stamps and subsidies for clean energy.

“The American people are counting on us to get this done and get it done quickly, and we are on target to do it,” Speaker Mike Johnson said on Wednesday morning at a news conference at the Capitol.

But even as they moved toward winning committee approval of the plan, House Republican leaders faced pushback in their own ranks that could delay or derail passage. Conservative lawmakers have argued the proposed cuts to Medicaid, which stopped short of an overhaul in an effort to protect vulnerable Republicans from political blowback, do not go far enough in restructuring and slashing costs of the program. They are unhappy that the largest reduction included — new work requirements for beneficiaries — would not take effect until 2029, putting off any savings until then, after the next presidential election.

And Republicans from high-tax states like New York were furious about a provision that would increase the limit on the state and local tax deduction to $30,000 from $10,000, a cap they regard as far too low and which was still being negotiated.

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