The Senate on Sunday moved toward a debate on the embattled Republican tax cuts and domestic policy bill, as G.O.P. leaders toiled to build enough support in their own ranks to push it through before a Fourth of July deadline set by President Trump.

Democrats who are unified against the measure protested its consideration by forcing Senate clerks to read the 940-page bill aloud, a maneuver that delayed the debate and was likely push any major votes to Monday. But even as the endgame drew nearer for the legislation, which would extend tax cuts first enacted in 2017 and exact steep cuts in Medicaid and nutrition assistance programs to help pay for them, continued to change.

The Senate official who enforces the chamber’s rules determined that two last-minute provisions — added on Saturday to benefit Alaska and Hawaii and help secure the vote of Senator Lisa Murkowski, Republican of Alaska — violate Senate rules and would likely need to be dropped from the bill, according to an aide.

Those provisions were crucial to winning the support of Ms. Murkowski on the initial procedural vote that narrowly passed late Saturday night on a 51-49 vote, with two Republicans and all Democrats opposed. It was uncertain whether she would still back the legislation if those sweeteners were dropped.

The ruling by the chamber official, the Senate parliamentarian, involved a special boost to the states’ Medicaid payment rates and one to the prices Medicare pays hospitals in those states for some medical services.

Adding to the uncertainty around the measure were new estimates by the nonpartisan Congressional Budget Office, which showed that it would pile at least $3.3 trillion to the already-bulging national debt over a decade, nearly $1 trillion more than the House-passed version. That could pose big problems for the measure in the House, which must give it final approval and where fiscal hawks have warned that the price tag of the measure must not rise.