The incident happened around 10.30pm on Wednesday, June 2 in the King’s Park area of the city.
The occupants of a white Seat Toledo drove up Castlemilk Road, stopped in front of a parked car, a black Audi Q7, and vandalised it.
A man came out his out to assess the damage to the car when the Seat Toledo turned round and attempted to drive towards him.
He was able to jump out the way of the vehicle and was uninjured.
Officers are treating the incident at attempted murder.
There is limited information on the occupants of the Seat, they are male and white.
The police do not know how many people were in the Seat and they are reviewing CCTV images for any further details.
Detective Constable Matt Potter said: “I am appealing to anyone who was in the area around the time of the incident.
"In particular, I would ask motorists and taxi-drivers with dash cams to check their footage as they may have captured an image of the Seat Toledo and the image could assist us in identifying those involved.
"Any small piece of information could prove vital and I would ask people to pass on what they know.”
Anyone with information is asked to call Police Scotland via 101, quoting incident number 3981 of the 2 June, 2021.
Alternatively, call Crimestoppers on 0800 555 111 where information can be given anonymously.