WASHINGTON — Eight days after blocking it, Senate Republicans have agreed to begin debate on a House bill that would overhaul the National Security Agency’s handling of American calling records while preserving other domestic surveillance provisions.

But that remarkable turnabout didn’t happen soon enough to prevent the laws governing the programs from expiring at midnight Sunday as Republican Sen. Rand Paul, a presidential contender, stood in the way of extending the program, angering his GOP colleagues and frustrating intelligence and law enforcement officials.

Now, the question is whether the Senate will pass a bill the House can live with. If so, the surveillance programs will resume, with some significant changes in how the phone records are handled. If not, they will remain dormant.

The Senate vote on the measure known as the USA Freedom Act can come no earlier than 1 a.m. Tuesday. Senate Republican aides said they expected some amendments, but no major revisions to the bill.

“Having gone past the brink, the Senate must now embrace the necessity of acting responsibly,” said Adam Schiff, the ranking Democrat on the House intelligence committee, in a statement after Sunday’s Senate vote.

The high-stakes drama played out as Congress debated the most significant changes prompted by the disclosures of Edward Snowden, the former NSA contractor who revealed the phone records collection and other main surveillance programs. With no deal reached in time, the NSA stopped collecting American phone records at 3:59 p.m. Sunday, officials said.

Intelligence officials publicly warned of danger, but were not deeply concerned with a lapse of a few days or weeks, given that the authorities remain available in pending investigations. What they most fear is a legislative impasse that could doom the programs permanently.