Road Policing Officers are now appealing for information following the incident.
Officers received a report that a 35-year-old man had been hit by a black Mercedes-Benz A Class car in Colinton Road in the city at around 11.35pm on Saturday.
Emergency services attended and the male pedestrian was taken by ambulance to Edinburgh Royal Infirmary.
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Medical staff describe his condition as critical.
The 20-year-old driver of the car was uninjured.
A section of the road was closed for around six hours to allow crash scene investigators to conduct their enquiries.
Sergeant Iain McIntyre said: "We believe the area would have been busy at the time and are keen to speak to anyone who may have witnessed the collision.
"In particular, we are keen to speak to the driver of a black cab, who was either dropping off, or picking up, passengers in Colinton Road at the time on the incident.
"We would also be keen to speak to those passengers as they may have information which could be vital to our enquiries.
"I would ask anyone with dash cam devices to check their footage as it could hold images, which could prove to be significant in our enquiries."
Anyone with information can call 101, quoting reference 4406 of Saturday, 30 July, 2022, or call Crimestoppers anonymously on 0800 555 111.