The young man (22) was part of a trio walking past a cash and carry on Maxwell Road when a woman started shouting and assaulting him.
The 22-year-old was left with injuries to his head and hand and he was taken to the Victoria Infirmary.
Staff at the hospital have described his condition as serious.
Police say the attack happened on new year’s day at 7.25am.
Detective constable Lyndsay Connor, of Gorbals police office, said: “Our initial inquiries have revealed that the man and woman responsible may have been involved on an altercation with another man at Maxwell Road, near to its junction with Herriet Street a short time before the serious assault took place.
“The man then left in a taxi, believed to be a white Skoda Octavia.
“This incident has not been reported to police and enquiries are being carried out to establish more details on this and to trace the victim and driver of the private hire taxi.”
Police have described the male attacker as aged between 10 and 21 years of age, between 5ft 6ins 5ft 8ins in height, of medium build with brown hair.
He wore grey jogging bottoms and no top and had scratches on his face.
The woman was described as white, aged between 20 and 22, also between 5ft 6ins and 5ft 8ins in height, of slim build with brown hair.
She was wearing a light-coloured top with a dark top underneath, shorts or a skirt and black shoes.