Police have confirmed they are treating the fire, which broke out at Ardencraig Drive, Castlemilk, at 10.20pm last Thursday, as "suspicious."
Detectives have so far discovered that the blaze was set on the ground floor.
It is understood fireraisers set alight a pile of rubbish left in the close of the three-storey building.
Now officers are appealing for anyone with information to get in touch.
The Evening Times reported how firefighters rescued the residents, including three children.
They were treated for smoke inhalation at the scene before being taken to Glasgow Victoria Infirmary.
A 28-year-old woman broke her wrist as she fled her flat.
Police said the investigation is yet to establish the exact cause of the fire, although officers are treating it as suspicious.
Officers are in the process of gathering CCTV images to find out more information.
Anyone with any information should contact officers at Cathcart Police Office on the non-emergency number, 101.
Alternatively, please call Crimestoppers on 0800 555 111.