WASHINGTON — Obscured by the furor over surveillance set off by the investigations into possible Trump campaign coordination with Russia during the election, a major debate over electronic spying that defies the usual partisan factions is quietly taking shape in Congress.

The debate centers on the National Security Agency’s incidental eavesdropping on Americans via its warrantless surveillance program, which spies on foreigners abroad whose communications pass through American phone and internet services. Its legal basis, the FISA Amendments Act, is set to expire at the end of 2017.

A bipartisan coalition of privacy-minded lawmakers has started to circulate draft legislation that would impose new limits on the government’s ability to use incidentally gathered information about Americans who are in contact with foreign targets.

Many of those lawmakers are veterans of a fight two years ago over the U.S.A. Freedom Act, a law that ended an N.S.A. program that gathered Americans’ calling logs in bulk. They won that fight against security hawks because the statute on which the program was based, part of the Patriot Act, was expiring and they were unwilling to extend it without ending the bulk collection.

The privacy advocates in Congress are using that same lesson this time around, hoping to leverage their colleagues’ concerns that the program will lapse if they fail to extend the law.

But the intelligence and law enforcement communities and their allies in Congress appear determined to extend the warrantless surveillance program law, Section 702 of the FISA Amendments Act, without changes. They are framing the debate as being about a program that is too important to be held hostage to any push for changes, lest gridlock kill it.

“This is a tool that is essential to the safety of this country,” the F.B.I. director, James B. Comey, told Congress at a hearing on Wednesday. “I did not say the same thing about the collection of telephone dialing information by the N.S.A. I think that’s a useful tool; 702 is an essential tool, and if it goes away, we’ll be less safe as a country. And I mean that.”

Mr. Comey also warned that one of the proposed changes — a new requirement that a warrant be obtained to search for Americans’ information in the surveillance repository — risked a failure to “connect dots” about potential threats.

But Rep. Ted Poe, R-Texas, sought to warn other lawmakers that Congress needed to impose a warrant requirement.

“Privacy is being betrayed in the name of national security,” Mr. Poe told congressional aides at an event last month.

There has already been some jostling over that idea. In 2014 and 2015, the House approved amendments to require warrants, but they died in negotiations with the Senate. When the idea came up again last year after the terrorist attack on a nightclub in Orlando, Fla., however, the House voted it down.

More broadly, two key members of the coalition that won privacy gains two years ago — Rep. John Conyers Jr. of Michigan, the ranking Democrat on the House Judiciary Committee, and Rep. Jim Sensenbrenner, R-Wisconsin — are discussing a draft bill, which is circulating among panel members.

A congressional staff member who described the draft on the condition of anonymity because it is not yet finalized or public said it incorporated a warrant requirement for American searches — with certain exceptions — and would:

* Restrict law enforcement from using information obtained or derived from warrantless surveillance except when investigating the most serious crimes, like murder.

* Reduce to three years from five years the time the government may retain unprocessed messages collected without a warrant.

* Codify a change that the N.S.A. and Foreign Intelligence Surveillance Court made to the program, ending so-called about collection of emails, from internet network switches, that talk about a foreigner being spied on but are not to or from that target.

* Require disclosures from the government that are more complete, including forcing the F.B.I. to say how often its agents ask for Americans’ information in national security cases, which it has declined to track.

The draft was part of a discussion at a Monday closed-door meeting convened by the Judiciary Committee chairman, Rep. Robert W. Goodlatte, R-Va., according to the congressional staff member. On Tuesday, Mr. Goodlatte said changes would be necessary.

“I know there are some people who want a clean reauthorization of Section 702,” he said.

“I don’t believe that is possible.”