A 21-year-old man was hit by a silver Mercedes E220 in Bolsover Street, in the city centre, at around 11.57pm.
The Mercedes was travelling towards the Brook Hill roundabout.
The collision occurred around 60 metres from the roundabout.
South Yorkshire Police said the man was taken to hospital with potentially life-changing injuries, where he remains in a critical condition.
Anyone with information should call South Yorkshire Police on 101, quoting incident number 1181 of December 16, 2017.