WASHINGTON — House leaders have agreed to permit a vote on tightening limits on when the F.B.I. may collect Americans’ internet browsing and search records during national security investigations, after negotiations over Memorial Day weekend between two California Democrats, Representatives Zoe Lofgren and Adam B. Schiff.
A vote on the proposal — an amendment to a peripatetic bill related to the Foreign Intelligence Surveillance Act, or FISA, that both the House and Senate have passed in different forms — is likely to come this week. If it passes, the bill would return to the Senate, extending its long-running consideration.
But by Tuesday night, there were multiple signs of political turbulence that raised new doubts about the bill’s future.
The text of the compromise amendment was not yet public, but congressional aides said that the proposal essentially limits to Americans the protections of a Senate proposal that would categorically ban the F.B.I. from using a court order for business records to collect internet browsing and search records.
The House is preparing to vote this week on the overall bill, which centers on extending three partly expired tools the F.B.I. uses to hunt for spies and terrorists.
It has also become a vehicle for broader changes to surveillance matters,