The incident took place at about 00:10 on Dalmellington Road in the Crookston area of the city.
The taxi driver's victim, a 24-year-old man, is in a critical condition in the Queen Elizabeth University Hospital.
The driver did not stop after the incident and drove off along Dalmellington Road towards Crookston Road.
The driver of the taxi has been described by police as of Asian appearance, aged 30-40, medium build, with short dark hair and wearing a pink top.
Det Insp Greig Wilkie said: "From our investigations so far, it would appear that the two men had left a party nearby and had asked the taxi driver to get a lift home, however, the driver, for whatever reason, refused to take them. It then appears that the injured man went on to the road which is when the taxi deliberately drove at him.
"The taxi didn't stop and drove off along Dalmellington Road, towards Crookston Road at Raeswood Road.
"On hearing a commotion outside, people from nearby houses came to assist the man until the ambulance arrived arrival of the ambulance. We are in the process of tracing them and others who were in the area, including the occupants of another black Hackney taxi who had been dropped off nearby just at the time of the incident, however, we have yet to trace the driver of that taxi and indeed another Hackney taxi driver who we know, were also in the area at the time."
He added: "This was a shocking incident, which has left a young man very seriously injured.
"I would appeal to the taxi driver who involved to come forward to police himself as soon as possible."
Police are looking for a Glasgow taxi driver who attempted to murder a man by driving his cab at him.