The debt limit deal negotiated by President Biden and Speaker McCarthy — one that includes spending cuts, new regulations on welfare programs, and expedited permitting for energy projects — faces its next hurdle Tuesday in the House Rules Committee, where key opponents of the bill hold power. 

Mr. McCarthy has called the bill “historic” and a win for conservatives, but some of his prominent Republican colleagues disagree. Two members of the conservative House Freedom Caucus — Congressmen Chip Roy and Ralph Norman — have called the legislation “not a good deal … with no substantive policy reforms” and “insanity,” respectively.

Congressman Thomas Massie — a libertarian — has said he is studying the bill but has drawn ire from his party’s leaders in the past for not staying in line. Messrs. Roy, Norman, and Massie all serve on the Rules Committee. Should those three Republicans and all four of the committee Democrats oppose passage, then it would be stalled.  

Mr. McCarthy may have to rely on Democrats on the Rules Committee and on the House floor if he hopes to pass the bill. A center-left caucus in the House, the New Democrat Coalition, publicly endorsed the bill Monday, buoying hopes that Mr. McCarthy can count on a substantial number of Democrats. 

Messrs. Biden and McCarthy “have achieved a bipartisan agreement that will save our country from default until 2025 and protect our nation from economic collapse,” the group wrote in a statement. There are 94 members of the New Democrat Coalition serving in the House. 

That a deal had been reached “in principle” was announced by Messrs. Biden and McCarthy late Saturday evening. The Fiscal Responsibility Act, as it is known, is 99 pages long and includes a number of modest changes to federal spending and regulations. 

The president and the speaker agreed that the debt limit must be pushed high enough so that this level of brinksmanship does not occur in the shadow of the 2024 elections. The bill would raise the debt limit by more than $2 trillion — pushing the next debt limit fight to January 2025 at the current pace of spending. 

The only two spending areas to see year-over-year increases are the Pentagon and veterans’ services. All other discretionary spending — from healthcare, education, and research to green energy investments — will be capped at a level that will result in hundreds of billions of dollars in savings over the next six years. 

Mr. McCarthy’s team also won a victory by slashing the budget of the Internal Revenue Service. In total, more than $20 billion will be slashed from the agency responsible for collecting taxes — a 25 percent cut to its total budget. 

Republicans took aim at the IRS early in this Congress, passing a bill that would rescind funding for the more than 80,000 new IRS agents who were hired as a result of the so-called Inflation Reduction Act last year. 

In total, just eight members of the Republican caucus had publicly opposed the debt limit as of Monday evening. 

The Biden administration agrees with the conservative House members that the deal does not represent a significant change in federal spending. “It’s flat,” one White House source told NBC News of the deal. “It’s a difference of about $1 billion. “In a divided government, we’re not going to get” the spending levels Democrats want.

Appeasing his right flank is a key priority for Mr. McCarthy. Congressman Eli Crane — a Freedom Caucus member who believed the original Republican debt limit proposal did not go far enough — has told CNN that he has had conversations with some of his Freedom Caucus colleagues about calling for a vote of no confidence in the speaker, which could lead to another seemingly endless voting process to either retain Mr. McCarthy or choose a new leader.

“It does come up from time to time,” Mr. Crane said of the possibility of removing Mr. McCarthy. “We look at all of the alternatives and contingency plans that could play out over the next two years.”