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.

The far-right group posted a video on YouTube showing members in the Al-Manar Islamic Centre in Cathays on 20 August.
The visit followed claims made in national newspapers in July that Shaykh Ali Hammuda, an Imam at the mosque, was a "radical" preacher, which he strenuously denied.
Ali Hammuda has been approached by BBC Wales to comment.
In the video, Britain First members can be seen entering the mosque and questioning a member, who told them the mosque rejected extremism.
The Muslim Council of Wales also said the "radical" claims were "bizarre and certainly unfounded".
In a statement, the council said the mosque was "a vocal and committed partner in both countering the ideology of the so-called Islamic State and the social factors that lead to radicalisation."
By entering the mosque Britain First members could have broken an injunction which banned members from entering any mosque in England and Wales.
South Wales Police confirmed a complaint had been received and "the matter is being investigated".
Bedfordshire Police, which secured the injunction, also confirmed it is investigating.
Footage of a street brawl which appears to involve a bus driver has been shared on social media.
[NEXT_CONCEPT]
Police are investigating a complaint from a Cardiff mosque which received an uninvited visit from Britain First.