Fire crews were called to Bishopbriggs Cultural Centre, in Auchinairn Road, at about 02:15.
Firefighters extinguished a blaze in a lock-up area. Some damage was caused to the building but no-one was injured.
Police Scotland said the fire was being treated as wilful fire-raising and an investigation was under way. They have also appealed for witnesses.
Bishopbriggs Cultural centre is listed as a mosque on the Glasgow Muslims webpage.
A Police Scotland spokeswoman said: "A joint Police Scotland and Scottish Fire and Rescue Service investigation is under way into the cause and circumstances of the fire.
"This is still at a very early stage but it is being treated as wilful fire-raising."
A spokesman for the Scottish Fire and rescue service said: "We received a 999 call shortly after 02:15 this morning, reporting a fire at a building in Auchinairn Road in Bishopbriggs.
"We immediately sent two appliances from Bishopbriggs and Springburn fire stations and crews reached the scene about seven minutes later.
"They found a small fire within a lock-up and two firefighters in breathing apparatus extinguished the fire."
The spokesman added: "Our crews left the scene at around 03:30 this morning, however, specialist officers later attended to assist Police Scotland."
Police are investigating a deliberate fire at a cultural centre that is used as a mosque by the Muslim community.