WASHINGTON—Key U.S. intelligence powers are due for renewal by Congress in March, setting the stage for a potential showdown over the broader framework of government surveillance after a recent watchdog report called out Federal Bureau of Investigation stumbles in seeking to wiretap a onetime Trump campaign adviser.

At issue are powers granted to U.S. intelligence agencies under the Foreign Intelligence Surveillance Act, initially written after Watergate-era spying abuses, and expanded under the Patriot Act, enacted after...