That means Republicans cannot spare another vote.

That’s where the increasingly toxic relationship between Mr. Trump and Senator Bob Corker of Tennessee comes into play. Mr. Corker is a senior Republican member on the Budget Committee who had already expressed misgivings about the tax plan and the potential for higher federal deficits. Losing his vote could be fatal. But Mr. Corker has very close relationships in the Senate, and many Republicans doubt he would kill the budget or tax plan simply to spite Mr. Trump.

“Defeating the budget is dishing more punishment on your colleagues than it is on the president,” said Neil Bradley, senior vice president at the chamber and a former top Republican official in the House.

Acutely aware of the tightrope he is walking, Senator Mitch McConnell of Kentucky, the majority leader, wrote an op-ed for NBC News encouraging Democrats interested in cutting taxes to put aside their issues with the president and join Republicans. It is unlikely to have much effect on the usually party-line budget vote, but Republicans know their job would be much easier if they could peel off just one or two centrist Democrats facing re-election next year.

Any significant rewrite of the tax code is extremely difficult — a political truth underscored by the fact that the last major overhaul was in 1986. Republicans concede that there are plenty of disagreements of the magnitude that could easily scuttle a proposal.

But they argue that Republicans are in such a politically perilous position because of their lack of accomplishment thus far that lawmakers — and the White House — are going to have to overlook their differences and come together on a plan. Otherwise they’ll be staring into the abyss.

“Members on both sides of the Capitol are going to have to be flexible and swallow features that may give them political indigestion in order to get a bill,” said Bob Stevenson, a Republican strategist and former senior Senate aide who was on Capitol Hill for the last tax overhaul. “But as Ben Franklin said, ‘You hang together or hang separately.’”

Mr. Bradley, the chamber official, said he agreed that in a more conventional political environment, existing divisions among Republicans could be sufficient to kill a tax bill. “But there is such a growing realization that failure is not an option that they will be able to overcome a lot of these policy and political conflicts,” he said.