The amendment is chiefly sponsored by Representative Justin Amash, Republican of Michigan, and Representative Zoe Lofgren, Democrat of California. It would substitute in the text of another bill, dubbed the USA Rights Act, which would extend Section 702 by only four years.

The bipartisan coalition backing overhaul efforts — which includes some of the most conservative and most liberal members of the House — say that change is necessary to uphold the meaning and substance of Fourth Amendment privacy rights in light of 21st-century communications technology and surveillance powers.

But the F.B.I. and the intelligence community have balked at that proposal, saying it would impede their efforts to protect the country to require warrants to query information the government already possesses. There are also lawmakers of both parties — backed by House leadership — who oppose the amendment.

Aides to Representative Devin Nunes, the California Republican who chairs the House Intelligence Committee, distributed a one-page sheet this week denouncing the amendment as imposing “unnecessarily severe requirements” that would endanger Americans.

Complicating matters, the base bill backed by Mr. Nunes contains a gesture toward a court-order requirement, too. It would apply only under narrow circumstances: if F.B.I. agents have already opened a criminal investigation into the American whose information they are searching for, and if the agents have no national-security rationale.

Representative Adam Schiff of California, the ranking Democrat on the Intelligence Committee, said the warrant requirement in the base bill would be sufficient to “prevent the database from being used as a general tool to gather evidence and introduce it in court in cases that have nothing to do with terrorism.”

But the base bill would still permit routine queries for Americans’ information without warrants. Its warrant requirement would not apply to national-security-related queries by a range of agencies, including the C.I.A., the N.S.A. and the F.B.I. Nor would it apply to F.B.I. queries when agents are merely pursuing tips about an American but do not yet have enough evidence of wrongdoing to open a criminal investigation.