A 29-year-old woman was struck by the bus on West Maitland Street, Haymarket, near Morrison Street on Sunday, December 5 at 9pm.
She was taken by ambulance to the Royal Infirmary of Edinburgh where medical staff have described her condition as critical but stable, according to Police Scotland.
No one from the bus was injured.
Members of the public came to the women’s aid.
Sergeant Iain McIntyre said: "The area was quite busy at the time and I'd like to thank those who came to the lady's aid and spoke to police, however, I'd still ask that anyone who witnessed the collision, or who may have caught the incident on their dash-cam footage, contact police.
"Please call officers on 101 quoting reference number 3356 of Sunday, 5 December, 2021."