The clip, filmed in Norwich city centre, shows two men coming to blows in front of a stationary bus at the corner of Prince of Wales Road and Upper King Street.
Both are seen grappling on the pavement before more punches are thrown.
Bus operator Stagecoach said a driver had been suspended. No arrests have been made.
In the video, the bus driver picks up what appears to be a badge from the middle of the street.
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He returns to the bus, and the two men continue to trade insults.
Andy Campbell, managing director of Stagecoach East, confirmed a bus driver had been suspended.
"We understand on-street video footage of part of the incident has subsequently been posted on social media", he said.
The company will launch its own investigation, and help police with their enquiries, he added.
Norfolk Police said they were called to Prince of Wales Road just before 16:00 BST on Wednesday.
They located one of the men when they arrived at the scene and were appealing for witnesses.
Footage of a street brawl which appears to involve a bus driver has been shared on social media.