The incident happened at around 1pm on Friday when the 47-year-old victim was involved in an altercation with three men who were in a car on Royston Road in Glasgow.
The driver of the white Ford Focus estate then deliberately drove towards the victim and struck him with the vehicle.
The Focus was driven off along Royston Road towards the Germiston area.
The victim was taken to Glasgow Royal Infirmary where he is currently being treated for a head injury and his condition is described as serious.
Officers are treating the incident as an attempted murder.
The driver of the car is described as being a white male, in his mid to late 20s, of thin build, short, and with fair hair, possibly shaven. This man did not have a West of Scotland accent. The two other occupants of the car are described as white.
Police have been checking CCTV in the area and carrying out extensive inquiries.
Detective Sergeant Nicola Kilbane said: "At the moment we have no clear motive for this attack on the 47 year-old victim.
"However we now have a good description of the vehicle involved which is a white Ford Focus estate model which has a 07 number plate.
"I would ask anyone who has any information about this vehicle or who may have seen it since lunchtime yesterday to get in touch.
"The vehicle itself would have sustained damage, and I would also appeal to any garages who have been asked about repairs to a white Ford Focus estate in the last 24 hours to get in touch."