Three men, aged 29, 30 and 47, were taken to Glasgow Royal Infirmary with serious injuries following the incident at around 5pm on Sunday.
It came on the evening of an Old Firm game which saw Celtic defeat Ranger 2-1 at Celtic Park.
Police said it was too early to say whether the incident was football related though it is understood that is one line of inquiry.
Police are treating the attack on the 47-year-old, who is in critical condition, as attempted murder and the attacks on the other two men as serious assaults.
Officers were called to the scene in Blackfriars Street in the Merchant City following reports of an altercation which appears to have involved weapons of some kind.
Police are appealing for information and asked anyone with footage of the incident to contact them.
Detective Inspector Peter Crombie of the Major Investigations Team, said: “We are currently going through CCTV and speaking to those who were in the area at the time to try to establish exactly what happened here.
“We are treating the attack on the 47 year-old man as attempted murder, and the attacks on the 29 and 30 year-old men as serious assaults.
“There may have been more people injured in this incident who did not seek medical treatment last night and we would appeal for them to come forward and speak to us.
“We also know that there were a number of people in the area who may have got caught up in it, or stopped to see what was going on. We would ask these people to check back and see if they have any mobile phone footage or images that can help us.
“If you were driving in the area, you may also have dash-cam footage that can help - either prior to the incident taking place or in the aftermath.”
The 30 year-old man remains in hospital in a stable condition while the 29-year-old has been released.
Anyone with any information is asked to contact Police via 101 quoting incident number 2923 of Sunday March 31, 2019 or Crimestoppers on 0800 555 111 where anonymity can be maintained.