Firefighters were called to the blaze in Main Street, Lennoxtown, on Saturday at around 3.30am.
The four unoccupied flats sit above shops.
Fire crews from Kirkintilloch, Bishopbriggs and Kilsyth tackled the blaze for up to four hours.
It is believed vandals went into the building and torched furniture in the flats. No one was injured in the blaze.
Police are investigating the incident and anyone with information is asked to contact them on 0141 532 4400.
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