It is reported that a blue BMW and a black VW were in a collision in Sandygate, at the junction of Quarry Hill Road, on Sunday, January 9.
The incident is said to have occurred at around 6.40pm.
South Yorkshire Police say the driver of the VW initially suffered minor injuries in the collision, but has since suffered a medical episode and has been left in a serious condition.
The driver of the BMW failed to stop at the scene but has since been identified.
Enquiries are ongoing.
Officers are now appealing to speak to anyone who may have information about what happened.
Anyone who witnessed the incident or who may have any relevant dashcam footages is asked to call 101, quoting incident number 633 of January 9.