Police officers believed that the vehicle contained a person wanted in connection with armed robberies in the south of England.
It happened at about 10am on Saturday morning on the westbound carriageway.
The car was spotted on the A64 near Scarborough at around 9.30am on and officers began to track the vehicle as it travelled westbound.
At around 10am, the black SUV hit the police car near Malton before flipping on to its side. Both vehicles were traveling in the same direction at the time.
The man, the only person in the car, was taken to hospital with suspected abdominal injuries but has since been discharged and is now in police custody having previously been arrested at the scene.
Two police officers were uninjured but attended hospital for a check-up.
Anyone who was travelling on the A64 at around 9.30am to 10am between Scarborough and Malton and witnessed the collision is asked to call North Yorkshire Police on 101 and pass information for incident 12190038235
The road was closed and reopened at around 4pm.