The incident happened after the man got off the Number 19 Lothian bus between 2.30pm and 2.50pm on Sunday, February 2.
The 23-year-old Polish man boarded the bus on Queensferry Street and was assaulted after he got off at the stop after the Crew Toll roundabout on Crew Road Gardens, leading to West Pilton Place.
It is believed that the attacker was also on the number 19 bus and got off at the same stop.
The attacker was wearing a light grey top and light blue jeans.
Detective Sergeant Lynsey Thomson from Police Scotland said: "This was an unprovoked attack which left the victim with a broken jaw.
"Following new information, we believe that passengers onboard the no. 19 and who left the bus at the same time may have witnessed this assault.
"Anyone with information that may assist with out inquiries is asked to contact police immediately."
Police are appealing to anyone with information that can assist their inquiries to contact Police Scotland on 101, or Crimestoppers in confidence on 0800 555 111.