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31 December 2014, 13:57 | Updated: 31 December 2014, 14:05
The 41-year-old woman and her young daughter were rescued by firefighters and taken to the Royal Alexandra Hospital in Paisley, where they are being treated for smoke inhalation.
Police believe the family was targeted deliberately and said the ''despicable act'' could have caused even more serious harm.
Detective Inspector David Wagstaff, of Paisley CID, said: ''We are treating this incident as attempted murder as it is believed that the fire was started deliberately outside the door of the flat.
''This was a despicable act which illustrates the culprit or culprits' total disregard for this family's safety. It could have left a mother and her young daughter with more serious injuries or indeed worse if it were not for the woman being awakened by her smoke alarm.
''We believe that this family was targeted deliberately and we are currently making inquiries as to the circumstances behind it.''
The Scottish Fire and Rescue Service (SFRS) used ladders to rescue the pair via a fourth-floor window and passed them into the care of ambulance crews.
Three engines from Renfrew and Govan community fire stations were sent to the scene after getting reports that the flat was on fire and people were trapped.
Six firefighters wearing breathing equipment put out the fire and carried out a search of the smoke-logged building.
The SFRS incident commander at the scene, station manager Kenny Barbour, said the crews ''worked tirelessly in arduous conditions to extinguish the fire and to ensure the safety of the occupants of the flats''.
Specialist officers from the SFRS investigation unit are involved in a joint probe with police to determine the cause of the incident.
Police want to speak to anyone who may have seen a person or persons hanging around Simons Crescent or the neighbouring streets at around 4am.
''If anyone has any information about this incident I would urge them to contact police as a matter of urgency,'' DI David Wagstaff said.
Anyone with information is asked to contact Paisley CID on 101 or Crimestoppers on 0800 555 111 where anonymity can be maintained.