Flames erupted from windows and onto the roof of William House in Marsh Lane, Walsall, during the fire on September 24.
It comes after 45 firefighters were called to an arson attack at former leatherworks Greatrex House in June last year, which was on the same street.
Nearby homes were evacuated and roads were closed as smoke poured over the area during the blaze in September.
West Midlands Fire Service's investigators said they believe the fire was started deliberately.
Evacuated residents were returned to their flats once the fire had been extinguished at about 9.40pm.
West Midlands Police has been asked to comment.