The blaze happened at a house in Gretna, Dumfries and Galloway, on March 26 after a 25-year-old man discovered the blaze at 1.30pm in the morning.
He was alone in the house in Torduff Road, but was able to extinguish the fire and did not suffer any injuries.
Detective Sergeant Hugh McCombe, Dumfries CID, has appealed to the public for information.
He said: “Thankfully the man was able to put out the fire and damage to the house was limited to the front door area. However, had he not woken up and tackled the fire, this could have been so much worse.
“He didn’t think to call the police or fire at the time as he had managed to put the fire out himself. However, on reflection, he called police to advise later in the week.
“We know from our joint investigation with Scottish Fire and Rescue that this fire was deliberately set but a motive has still to be established.
“Torduff Road is a residential area and is near to Annan Rd in Gretna. I am appealing to anyone who may have seen someone hanging about the area they did not recognise, or who may have any information about the attack to come forward to police.”