The 32-year-old pedestrian had flagged the car down on Barnsley Road in the early hours of Sunday thinking it was a taxi.
The driver initially stopped and the pedestrian tried to get into the vehicle.
The driver then began to make off and the side of the car hit the pedestrian, causing him to fall and sustain a serious head injury.
He remains in hospital in a stable but serious condition.
PC Ray Terry, of the Safer Roads Team, said: “From our initial enquiries, we don’t believe the driver realised he or she was in collision with the 32-year-old male.
“We would like to speak to the driver as part of the investigation but we don’t believe they are at fault and they aren’t any in trouble.
“I would appeal for anyone who witnessed this incident to please get in touch with police, and I would also ask the driver of the vehicle involved to also contact us."
Call the Safer Roads Team on 101.