Two groups of youths were involved in the incident in Buchanan Street at about 16:10 on Friday.
An 18-year-old, and a boy aged 15, sustained stab wounds and were treated in hospital. The 18-year-old remains in Glasgow Royal Infirmary.
Officers believe a third teenager was injured but left the scene.
Det Sgt Colin MacRitchie said: "What we know is that a group of six to seven male youths were in Buchanan St, between West George Street and Bath St, when three of the group became involved in an altercation with at least one other man in the street.
"A number of his friends came to his aid and it is then that the fight took place.
"From CCTV and our inquiries, we understand that one man sustained an eye injury during the incident but he left before police arrived. It is essential that we trace him both to find out if he is OK and also what information he has about the assault."
He added: "We are also looking for three suspects who made off west in Bath Street following the attack. They are all described as white and in their late teens, early 20s."
Buchanan Street was busy at the time of the fight and some of the violence was captured on mobile phones.
Det Sgt MacRitchie asked anyone with video or stills to get in touch.
He said: "We know from social media and from speaking to witnesses that some of it was captured on mobile phones.
"If you have any footage at all, please let the police know. Police are also checking CCTV in the area to identify the third injured man and the suspects."
Police in Glasgow have confirmed they are searching for a third youth they believe was injured during a fight in the centre of the city.