The 52-year-old suffered serious head injuries after being attacked in the busy street outside a Tesco Express store.
Police raced to the store in Argyle Street last night after the alarm was raised.
Detectives from the major investigations unit revealed the two men were involved in an argument inside the store at around 9.45pm.
They then went outside and the 52-year-old was attacked.
He was taken to the Southern General Hospital, where medical staff today described his condition as "critical".
Officers believe a man and a woman who were in the store at the time may hold vital clues about the attack.
Detective Inspector Fil Capaldi, from Police Scotland Major Investigation Team, urged the two people to come forward.
He said today: "Argyle Street would still have been busy at the time of the incident with people making their way to pubs and restaurants.
"I would urge anyone who witnessed this incident take place outside Tesco, or indeed the disturbance within the shop, to come forward to police as a matter or urgency.
"I am particularly interested in speaking to a man and woman who were in the Tesco store at the time of the incident."
Anyone with any information is asked to contact Police Scotland on 101.