The pedestrian, in her 20s, was struck on the Dunchurch Road at 12:20 GMT, police said.
The car that hit her was being driven on a 999 callout to the A45 at Stretton-on-Dunsmore.
West Midlands Ambulance Service said the woman was anaesthetised at the scene and taken to University Hospital Coventry.
The IPCC said it had sent investigators to the scene of the crash at the junction of Dunchurch Road and Overslade Lane.
Eyewitness David Williams said: "A container lorry was coming along and behind that I could see there was a police car - blue flashing lights, siren on.
"I looked because it was coming quite fast. The road was in essence blocked by the slowing container lorry and the police car proceeded to pass him on the other side of the carriageway."
A woman has suffered serious head injuries after being hit by a police car in Rugby.