The 69-year-old man was struck when the bus turned right on to the West Approach Road, from Lothian Road, at about 11.10am on Tuesday morning.
He suffered a suspected head injury and was taken to Edinburgh Royal Infirmary, where he is still being treated.
Police are now asking anyone who may have been in the area at the time, and who saw what happened, to come forward.
Inspector Richard Latto said: “We know there were a lot of people on foot and in cars in the Lothian Road area, so we would ask anyone who may have not yet spoken to police to contact us on 0131-311 3131.”
Alternatively, anyone with any information should call Crimestoppers anonymously on 0800 555 111.