Trump on Monday told John Thune to keep the Senate in session past the Aug. 7 break until it passes the SAVE America Act or scraps the 60-vote filibuster.
53 Republicans hold the chamber, leaving the party short of the votes needed to advance the bill, change Senate rules or use a budget-reconciliation fallback tied to a House-approved $10 billion framework.
Thune said he would support canceling recess only if Republicans can show a path to victory, noting the Senate has already voted on the measure five times without breaking the impasse.
Mike Lee and John Kennedy have intensified pressure on Thune, but Thom Tillis and John Cornyn said most GOP senators back the leader's view that staying in Washington will not change the math.
The clash threatens Thune's final pre-recess agenda on funding, crypto and Russia sanctions while Republicans head into competitive midterm races amid high costs, surging gas prices and an unpopular Iran war.