WASHINGTON—A bipartisan group of U.S. lawmakers unveiled legislation on Wednesday that would overhaul aspects of the National Security Agency’s warrantless internet surveillance program in an effort to install additional privacy protections.

The bill, which will be formally introduced as soon as Friday, is likely to revive debate in Washington over the balance between security and privacy, amid concerns among some lawmakers in both parties that the U.S. government may be too eager to spy on its own citizens.

The legislation, written by the House of Representatives Judiciary Committee, is seen by civil liberties groups as the best chance in Congress to reform the law, known as Section 702 of the Foreign Intelligence Surveillance Act, before its expiration on Dec. 31.

Senior U.S. intelligence officials consider Section 702 to be among the most vital tools they have to thwart threats to national security and American allies.

It allows U.S. intelligence agencies to eavesdrop on and store vast amounts of digital communications from foreign suspects living outside the United States.

But the program, classified details of which were exposed in 2013 by former NSA contractor Edward Snowden, also incidentally scoops up communications of Americans, including if they communicate with a foreign target living overseas. Those communications can then be subject to searches without a warrant by the Federal Bureau of Investigation.

A discussion draft of the legislation, a copy of which was seen by Reuters, partially restricts the FBI’s ability to access American data collected under Section 702 by requiring the agency to obtain a warrant when seeking evidence of a crime.

That limit would not apply, however, to requests of data that involve counterterrorism or counter-espionage.

The narrower restriction on what some have called a “backdoor search loophole” has disappointed some civil liberties groups. Several organizations sent a letter this week saying they would not support legislation that did not require a warrant for all queries of American data collected under Section 702.