Even before the House passed the sweeping bill carrying President Trump’s domestic policy agenda, Senate Republicans made it clear that they hoped to make major changes to the legislation before the G.O.P. was done muscling it through Congress.

Several have wanted to pare back the cuts to Medicaid, the health care program for the poor, that House Republicans envisioned in the version of the legislation that they approved late last month. A handful have sought to salvage tax credits incentivizing clean energy projects that the House measure would repeal. Many have pushed to grant companies prized tax breaks for the long run, not just for a few years, as their colleagues across the Capitol opted to do.

The problem senators face is that each of these changes would be expensive. At $2.4 trillion, the cost of the legislation that barely passed the House is already huge. So Senate Republicans are now hunting for ways to save money, a hazardous task that could involve shaving the ambitions of their colleagues in the House or in the White House.

On the chopping block are some of Mr. Trump’s favorite parts of the bill, like not taxing overtime. Republican lawmakers have long been skeptical of some of the president’s tax ideas, with the view that the populist policies will not spur the economy like traditional supply-side conservatism can.

“I think it all comes down to what we’ve got to pay for,” Senator Thom Tillis, Republican of North Carolina, said. “At the end of the day, we’ve got to pay for pro-growth policies.”

The debate is in some ways a classic one on Capitol Hill, where throughout history and without regard to political party, senators have been reluctant to defer to their colleagues in the House, and vice versa.