The Daily Record reports that an explosive ‘petrol bomb’ like device was thrown at a car in the driveway of Mr Lawwell’s home on Peel Road at around 1am on May 19, while the family were asleep.
Celtic FC said the family are “extremely shaken and shocked” by the blaze, which caused “significant damage”, but they are all safe.
A Celtic spokesman said: “We can confirm that significant damage has been caused to Peter Lawwell’s house and vehicles there, following an explosion and fire early this morning, forcing the family to leave the property.
“Clearly, Peter’s family are extremely shaken and shocked by these terrible events, but thankfully all are safe
“We understand that Police Scotland are currently undertaking a criminal investigation.
“Peter and his family will of course receive the full support and care of everyone at the club.”
Police officers and eight appliances from the Scottish Fire and Rescue Service (SFRS) attended the scene.
The SFRS said that the fire spread to a number of cars as well as the two-storey home.
Firefighters extinguished the blaze but some remained at the scene at 7am, police road closures were still in place at that time.
The spokesperson said: “We were alerted at 1.02am on Wednesday, May 19 to reports of a car on fire outside a dwelling on Peel Road, Thorntonhall, Glasgow.
“Operations Control mobilised eight appliances to the scene to extinguish the fire, which was affecting a number of cars as well as the detached two-storey property.
“There are no reported casualties.”
A Police Scotland spokesperson said: “Around 1am on Wednesday, May 19, officers were called to a report of vehicles on fire outside a house on Peel Road, Thorntonhall.
“No one was injured and officers are currently working with the Scottish Fire and Rescue Service to establish the cause of the fire.”