The incident happened about 1pm on Friday in Royston Road in the East End.
The 47-year-old victim was seen talking to three men in a white Ford Focus estate when an argument broke out.
The driver then deliberately drove towards the victim knocking him to the ground.
The car sped off along Royston Road towards the Germiston area.
A member of the public phoned the emergency services and the man was taken by ambulance to Glasgow Royal Infirmary. He is being treated for a head injury which medical staff described as "serious".
Police say they are treating the attack as attempted murder and have appealed for witnesses.
The driver of the car, who did not have a Glasgow accent, is described as a white male aged in his mid to late 20s who is thin and short with fair hair which is possibly shaven.
The other two people in the car are said to be white but police say there is no further description. Officers have been carrying out extensive inquiries and are checking CCTV to trace the men responsible.
Detective sergeant Nicola Kilbane said: "At the moment we have no clear motive for the attack.
"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 and I would ask anyone who has any information about this vehicle to get in touch.
"The vehicle would have sustained damage and I also appeal to any garages who have been asked about repairs to a white Ford Focus Estate to get in touch."
Anyone with information is asked to contact Baird Street CID on 101.