Police said "several men" were involved in the violence outside the supermarket, in Mill Road, Hamilton.
Emergency services were called to the scene at around 8.15am and all of the men involved were taken to Wishaw General Hospital.
Two were treated for what is believed to be stab wounds.
A large section of the busy road was cordoned off and police stood guard at the scene.
Mill Road remained closed and local diversions were put in place.
A spokeswoman for Strathclyde Police said: "Several men were injured in the incident and two received serious injuries."