The incident happened in Penistone Road at 6.20pm on Saturday.
A marked Astra police car on routine patrol hit a 70-year-old man who died at the scene, a South Yorkshire Police spokesman said.
Officers are supporting the family of the man, police said, and the force offered its ‘sincere condolences’ to his family and friends.
The crash happened near KFC at the junction with Bradfield Road and the area was closed off last night as specialist staff investigated.
Anyone who witnessed what happened, or who can help police with the investigation, is asked to call 101 quoting incident 1021 of January 7.