Firefighters were called out to Westhouse Drive, Mayfield, at around 2.15am this morning after the fire was reported.
Four flats within the building were evacuated while the fire service extinguished the blaze.
Police said the fires were “suspicious” and are now investigating the incident.
Inspector Michelle Ritchie said: “This was an incredibly reckless act in the early hours of the morning, which resulted in four flats being evacuated. Luckily no one was injured and the fire service were able to extinguish the fires quickly.
“Anyone who has any information that can assist with our enquiries, or who noticed any suspicious activity in around Westhouses Drive last night and this morning, should contact police immediately on 101.”