Among those on the fence are Senator James Lankford, Republican of Oklahoma, who has expressed concerns about the bill’s impact on the budget deficit but favors ending the 1954 ban. In a possible sign of the horse trading to come to try to secure votes, a spokesman for Mr. Lankford said on Sunday that the senator was working to insert language into the Senate bill to roll back the ban, and believed it had a good chance of being included.

If the bill passes the Senate, lawmakers will still need to resolve key differences between the House and Senate bills, including whether to make the tax cuts for individuals permanent, as the House bill does, or temporary, as in the Senate legislation. Leaders will also need to decide what to do about popular tax breaks, like the mortgage interest and state and local tax deductions, which each bill treats differently.

With time running out for Republicans to deliver a major legislative victory after nearly a year of stalemate on the party’s top agenda items, lawmakers appear poised to agree to last-minute changes and tweaks to try to ensure the bill’s passage so it can be delivered to President Trump by Christmas. Tax bill negotiations are expected to kick into high gear on Monday, as lawmakers return to Washington with just a handful of legislative days left and big issues to contend with, including the need to pass a funding measure to keep the government open beyond Dec. 8 and action to protect the young undocumented immigrants known as Dreamers.