The alarm was received shortly before 3 a.m. Thursday, according to WLKY-TV in Louisville.
Fire officials say the blaze began on the first floor of an apartment and spread upward. Firefighters reported heavy flames when they arrived. The fire has been contained.
The extent of injuries among people sent to hospitals wasn't immediately clear, but officials told the station at least one of the injuries appeared serious at the scene. There was initial concern that a person was missing, but officials later said that person is among those taken to hospitals.