The fire that broke out on 16 July at the Nasfat Islamic Centre building on Droylsden Road, Newton Heath, was being treated as a "hate crime", Greater Manchester Police said.
A force spokesman said a man had approached the premises that evening and forced open a window before placing an "unknown accelerant" inside.
The arrested 23-year-old has been released under investigation.
The fire, which was discovered at about 23:40 BST by officers on patrol, damaged a prayer room and three classrooms.
Det Ch Insp Paul Walker said officers were "still working hard to establish the full circumstances around the arson attack".
A man has been arrested on suspicion of arson over an attack on a mosque.