WASHINGTON — Senate Republicans introduced a measure Tuesday night that would curtail future presidential emergency declarations in a last-ditch effort to save President Trump’s current border wall emergency from an embarrassing rejection by senators charging constitutional overreach.

Four Republican senators have publicly said they will support a House-passed resolution of disapproval when it comes up for final passage in the Senate on Thursday. That would be just enough to ensure passage of a measure overturning the national emergency that the president declared to secure border wall funding over Congress’s objection.

But that near certainty appeared to shift on Tuesday as Vice President Mike Pence pressed Senate Republicans at a closed-door meeting before their weekly policy lunch. A measure sponsored by Senator Mike Lee, Republican of Utah, and supported by more than a dozen Republican senators, would curtail the president’s powers under the National Emergencies Act of 1976, requiring a congressional vote of approval for any new emergency declaration after 30 days.

And one of the Republicans who said he favored the resolution of disapproval, Senator Thom Tillis of North Carolina, appeared to be wavering, according to multiple people at the policy lunch. Without him — and barring any other defections — the resolution would fail to reach the president’s desk.