The incident happened at around 11.45pm on Friday at a bus shelter outside Bar Burrito on Hope Street.
The 60-year-old male victim sustained a serious injury after being assaulted by another male, and falling to the ground.
He was taken to Queen Elizabeth University Hospital where he remains in a critical condition.
The suspect is described as a white man in his forties, around 5ft 6ins tall, of medium build with short dark hair.
He spoke to witnesses nearby before walking away towards Gordon Street. He was wearing a dark three quarter length jacket, light coloured blue jeans and grey hat.
He had a dark holdall, a white polythene bag and a brown satchel.
Detective Sergeant Gordon Walker of Stewart Street CID said: “This was despicable behaviour which has resulted in the serious injury of a man.
"Enquiries remain ongoing but we are treating this as an attempted murder and we are working to establish the circumstances leading up to this incident, and to identify the man responsible.
“There would have been a number of people in the area around the time of the incident both on the street and in vehicles and taxis. I am appealing to anyone who saw what happened to contact police as soon as possible.
“I would also encourage anyone who was driving in the area to please check any dashcam footage for anything which might be useful to our investigation.”
Anyone with information should contact police on 101 quoting incident number 4236 of 8 October, or alternatively contact Crimestoppers on 0800 555 111.