Emergency services were called just before midnight on Sunday, June 27, when a member of the public found the man lying injured on Great Western Road.
The man was found near the Esquire House bar in Anniesland.
The 55-year-old was taken to the Queen Elizabeth University Hospital for treatment in a critical condition.
However, medical staff have now described the injured man as being stable.
Following the incident, officers are appealing for more information – especially from three youths seen on CCTV at the time.
Officers have said it is ‘imperative’ that they find out how the man sustained the injury.
Detective Constable Ross Thomson said: “At this stage the full circumstances are unknown however due to the fact the man was found with a serious head injury, it is imperative we ascertain how this happened.“The incident is believed to have occurred at some point between 11.30pm and 11.55pm and I would appeal to anyone who either stopped to assist the man, or was passing by, to please come forward.
“I am keen to speak with three youths captured on CCTV in the area around the time.
“I would also urge any motorists with dashcams who were travelling along Great Western Road at around that time to check their footage in case they have managed to capture anything of significance.
“Anyone with information is asked to contact Police Scotland through 101, quoting incident number 0210 of Monday June 28.”