The incident happened at around 1pm on Saturday, June 3 at the junction of Mythop Road and Chain Lane.
Officers say the cyclist, a 45-year-old man, was turning right onto Chain Lane, from Mythop Road, when he collided with a white Audi A1. The driver stopped at the scene but no details were exchanged.
The cyclist later attended Blackpool Victoria Hospital.
Police would now like to trace the male driver of the Audi, along with anybody else who saw the incident.
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PC Ian Byrne, of the Blackpool Roads Policing Unit, said: “We would like to trace the driver of the Audi, who stopped at the scene, but did not leave any details.
"If you believe you are this man, or you know who he may be, we would urge you to contact us as soon as possible. Similarly anybody who saw the incident and has not yet been spoken to is also asked to make contact with us so we can piece together exactly what occurred.”
Anybody with information is asked to email [email protected], quoting log number 1279 of June 3.