WASHINGTON — After vigorous debate and intense last-minute pressure by Republican leaders, the Senate on Saturday rejected legislation that would curb the federal government’s bulk collection of phone records.

With the death of that measure — passed overwhelmingly in the House this month — senators scrambled but failed to pass a short-term measure to keep the program from going dark when it expires June 1. The disarray in Congress appeared to significantly increase the chances that the government will lose systematic access to newly created calling records by Americans, at least temporarily, after June 1.

“This is a high-threat period,” said Senator Mitch McConnell of Kentucky, the majority leader, who was stymied in his efforts to extend the program even for a few days by the junior senator for his state, Rand Paul.

A senior administration official said Saturday that the “wind-down process has begun” on the surveillance program, and that the administration did not file an application with the Foreign Intelligence Surveillance Court on Friday to continue collecting the data. The White House has long said that the administration would not seek to continue the program if the legal authorities expired. Aspects of the program could be reactivated as allowed under new legislation if Congress acts before the deadline.