The Audi Q7 was doused in a flammable accelerant whilst Miller was visiting close family at a property in South Lanarkshire.
A Police Scotland spokesperson said: "Around 12.05am on Saturday, January 8, officers were called after an attempt was made to set alight a vehicle in a car park in Hamilton.
“It appears at this time the car was not damaged and no one injured.
“Enquiries are continuing.”
According to the Daily Record, the 42-year-old was temporarily staying with his wife Laura, 33, at his mother-in-law's home when the arson attempt took place on January 8.
The windscreen of the £62,000 motor was doused leaving significant damage to the car, although the suspects fled after being disturbed.
A source said speaking to the newspaper: “Kenny says he has no idea why this happened or what’s prompted it, the family are totally in the dark.
“He tries to keep his professional and family lives as separate as possible.
“They’ve got a lot going on just now with a new baby and Kenny’s new job.
“It’s shocking that this should happen to them.”
The coach has been working with Falkirk for just over a month, joining the club in December as head coach Martin Rennie’s number two after returning from a stint in Australia.