South Yorkshire Police has issued an appeal for witnesses following the incident at around 1.36am this morning (Sunday).
The force said it was reported that a black Vauxhall Insignia was in collision with a pedestrian on Regent Street in the city centre but did not stop at the scene. It then continued travelling in the direction of Portobello, then Brook Hill roundabout.
The pedestrian, a 58-year-old man, was found a short time later by members of the public and was taken to hospital, where he remains in a critical but stable condition.
Acting Sgt Rod McEnery, who is leading the investigation, said: “Officers are carrying out extensive enquiries to locate the driver of the vehicle so we can ascertain exactly what happened in the early hours of this morning.
“It’s vital that we hear from anyone who might have seen the vehicle prior to the collision, or afterwards, so we can piece together exactly what has happened.
“If you have any information, or saw anything that might help our investigation, I urge you to please contact us immediately. You can call us on 101, or submit information online using our online portal or live chat.”