The victim was walking outside a cash and carry on Maxwell Road, Pollokshields, with two friends when he was approached at about 7.30am on New Year's Day.
He is being treated for head and hand injuries at the Victoria Infirmary where doctors described his condition as serious. Police believe the attackers may have been involved in another altercation shortly before.
The man is described as white, aged 19-21, of medium build with short brown hair and scratches on his face. He wore grey jogging trousers and no top. The female suspect is white, aged 20-22, of slim build with brown hair. She was wearing a light-coloured top with a dark top underneath, a skirt or shorts and black shoes.
Detective Constable Lyndsay Connor from Police Scotland said: "Our initial inquiries 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.
"The man then left in a taxi believed to be a white Skoda Octavia. This incident has not been reported to police and inquiries are being carried out to establish more details on this and to trace the victim and driver of the private hire taxi, although I would ask them to come forward to police."
Police are studying CCTV and urged witnesses to call them on 101.