The man in his 70s was taken to Leeds General Infirmary with serious injuries after being hit by the car, which did not stop at the scene.
Witnesses are being sought to the incident at 11.55am in Longcauseway as the search continues for the driver.
A police spokesman said: "The car that was involved in the collision left the scene. It's believed to be a black Audi A3."
Anyone who saw the collision, or has any information about the car involved, is asked to contact PC 5894 Felicity Davis at SRANS West on 101, quoting log 556 of January 25.