The assaults happened in a car park at the Greenfield Football Centre in Duror Street, in the east end of the city, at about 20:20 on Saturday.
Police said the victims had been stabbed by a group of males in an unprovoked attack that “could have left them dead”.
Three men, two aged 21 and one aged 19, were taken to hospital.
Police are treating the incidents as two attempted murders and a serious assault.
All three men were taken to the Glasgow Royal Infirmary to be treated for serious stab wounds.
Officers have been carrying out extensive inquiries and are checking CCTV images.
Det Insp Stephen Greenshields said: “This has been an unprovoked attack on a group of young people which could have left them dead.
“The injuries inflicted were severe and extremely serious. It is unacceptable that people think they can behave in such a way and think they will get away with it. Let me assure you, they will not. A team of officers is working to establish the identities of the men responsible for this crime.
“I would appeal to members of the public who were in the area at the time who may have witnessed something to contact us. Any small piece of information could assist us as we piece everything together and find those responsible.”
Additional patrols will be in the area to provide reassurance to the local community and anyone with concerns can approach these officers.
