WASHINGTON—The Federal Bureau of Investigation said Thursday it will notify state officials when local election systems are believed to have been breached by hackers, a pivot in policy that comes after criticism that the FBI wasn’t doing enough to inform states of election threats, according to people familiar with the matter.

The FBI’s previous policy stated that it notified the direct victims of cyberattacks, such as the counties that own and operate election equipment, but wouldn’t necessarily share that information with...