The incident at around 3.20am on Sunday occurred as the trio, aged between 30 and 47, were walking together in St Vincent Street near Buchanan Street, which was busy with people leaving bars and clubs.
The older victim suffered a severe ear injury and is undergoing surgery at Glasgow Royal Infirmary. The two others have been discharged after treatment.
Detective Constable Stuart Nutt at the Glasgow City Centre CID said: "It is thought that the group of men responsible had been walking eastbound along St Vincent Street prior to carrying out this vicious attack. The area was particularly busy at the time of the incident with people leaving pubs and clubs in the area."
In a separate incident in Edinburgh, two men aged 24 and 30 and a 23-year-old woman who had been out drinking together were assaulted when one of the men bumped into another male in Frederick Street.
In the ensuring fight, the woman was knocked to the ground and sustained a fractured jaw.
Detective Constable Lynne Baxter said it was a "mindless act of violence."
Meanwhile, police have released CCTV images of a man they wish to trace in connection with an unprovoked assault on a man who was left seriously injured on December 29, 2012 in Silk Nightclub in the capital's Kings Stables Road. They are following a positive line of inquiry.