The incident happened in Hull in the early hours of Wednesday, April 3.
Police received a report that items had been stolen from a garden in the village of Sproatley by suspects driving a white Citroen van at about 1am.
Officers found the van in Hull and pursued the suspects.
The driver of the van reversed the car into a police car and damaged a second police car on Greenwich Avenue.
Two officers were injured in the pursuit.
The officers needed hospital treatment for minor injuries and are receiving support.
All three men are currently in police custody on suspicion of theft and dangerous driving.
Investigations are ongoing.
Chief Inspector James Glansfield said: “My officers come to work to make sure the communities we serve are safe. They do not expect to come to work and end up being injured.
“We do not tolerate any kind of attacks on police officers and we take such incidents very seriously.”