The 52-year-old victim suffered serious head injuries after being attacked in the busy street outside the Tesco store.
As reported in late editions of yesterday's Evening Times, a vicious argument broke out between two men.
Minutes later, they went outside and the 52-year-old was attacked.
Police raced to the Tesco Express store in Argyle Street, at around 9.45pm on Sunday, after the alarm was raised.
The victim was taken to the Southern General Hospital, where medical staff described his condition as "critical".
Detective Inspector Fil Capaldi, from Police Scotland Major Investigation Team, urged two people in the shop to come forward.
He said: "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 outside Tesco, or indeed the disturbance within the shop, to come forward 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."