A man is being treated in hospital for serious injuries following the incident in the Gorbals area.
Two men were seen fighting at about 8pm outside high-rise flats at Waddell Court on Sunday at around 8pm.
Emergency services were called and both were admitted to hospital.
A third man fled before police and ambulance crews arrived.
Detective Constable Claire Anderson, of Gorbals CID, said: "We are still trying to establish the exact circumstances of this incident and also trace the third man involved.
"After viewing CCTV of the area we know that a man and woman walked past the on-going fight and it is vital that they contact police as they could provide crucial information."
The first man was white, aged 50, had blond hair and was wearing a grey tracksuit. He is currently being treated at the Victoria Hospital suffering from serious injuries. Staff described his condition as stable.
The second man was white, aged 34, with a large scar on his face and wearing a dark hooded top. He was treated for his injuries at the Royal Infirmary before being released.
The third was younger than the other two and wearing a dark hooded top. He fled on a bike before ambulance crews arrived.