The 22 year-old was left with serious injuries after being assaulted in Pollokshields, in the city's South Side.
Witnesses said police were seen examining a blood-spattered car at the scene in Maxwell Road.
The man was walking with two friends at around 7.25am when they were approached by a woman, aged 20-22, outside a cash and carry. The woman shouted and assaulted the victim and she was then joined by a man, aged 19-21, who also attacked him.
Emergency services who rushed to the scene found the man lying on the road with head and hand injuries. He was taken to the Victoria Infirmary where his condition was described as "serious".
Police believe the attackers were involved in an earlier dispute with another man on Maxwell Road, near Herriet Street, before the assault took place. The man then left in a private hire taxi.
Police are trying to trace the driver of the white Skoda Octavia.
The male attacker had no top on and had scratches on his face.
Detective Constable Lyndsay Connor said: "Our initial enquiries have revealed that the man and woman responsible may have been involved in an altercation with another man at Maxwell Road, near to its junction with Herriet Street a short time before the serious assault took place.
"Police are continuing to study CCTV in the local area, and I would urge anyone with information that may help police enquiries, or who may recognise the description of the two suspects to contact 101, the non-emergency number for Police Scotland."