The vehicles crashed into each other in New Bedford Road in the town centre at around 4.10pm.
Emergency services had to free two people who were trapped after the crash, Luton Today reported.
The man in a critical condition is in his 40s but the ages of the other people who were injured are not clear, Bedfordshire Police said.
The two buses involved in the collision were operated by Grant Palmer and Centrebus.
A spokesman for Grant Palmer said the driver of their vehicle had been injured in the crash, but could not confirm if he was the man in a critical condition.
No arrests have been made.
We are aware of a bus collision in Luton resulting in a small number of injuries.
Please help us by avoiding our A&E for non life-threatening illness while we assist those who need urgent help.
— Luton and Dunstable University Hospital (@LandDHospital) April 10, 2018
The crash happened near the Galaxy centre, a complex which includes a cinema, bowling alley and arcade.
An air ambulance joined emergency crews before the injured were taken to nearby hospitals.
Luton and Dunstable Hospital issued a plea for people with non life-threatening injuries to avoid going to A&E as they treat the injured passengers.
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The section of New Bedford Road where the crash happened is closed while the two buses are removed.
Motorists have been urged by emergency services to avoid the area.