Police said the crash, involving a black Audi A1, happened at about 6.10pm in Gipton Wood Road yesterday (Monday).
The car was travelling on Easterly Road, towards Oakwood Lane, and had turned left onto Gipton Wood Road.
It then collided with the 35-year-old man while he was crossing the road.
West Yorkshire Police has today (Tuesday) launched an appeal for witnesses after the vehicle drove off from the scene following the crash.
The pedestrian was taken to hospital where he was today described as being in a "critical condition".
Officers today said it is suspected that the vehicle involved could have been stolen during a burglary at a home in Amberton Road, Gipton, earlier that day.
Detective Inspector James Entwistle, of Leeds CID, said: “We are treating this incident very seriously and conducting extensive enquiries into the circumstances around it.
“I would appeal for anyone who witnessed the collision or saw the vehicle driving in the area prior to the collision to contact police.
“I am also keen to hear from anyone who has seen the car since and from anyone who witnessed any suspicious activity on Amberton Road which may be connected to the burglary.”
The Audi carried the registration plate 'OV15 WDU', according to police.
Anyone with information is asked to contact Leeds CID on 101, quoting crime reference 13180516787.
Information can also be passed anonymously to independent charity Crimestoppers on 0800 555 111.