Fox Sports Australia is reporting Bairstow headbutted Bancroft, while other versions suggest it was more low key and a case of tomfoolery gone awry.
Cricket Australia (CA) refused to comment.
AAP understands Bancroft has discussed the matter with team management, who are satisfied he did no wrong in the scrap.
"At close of play in Brisbane today, we were made aware of allegations of an incident in Perth four weeks ago," an ECB spokesperson said.
"There has been no report of any incident from the venue, security or police and there was no injury reported.
"Following an initial conversation with Jonny Bairstow tonight we understand the context and will follow up with England players and management after the Brisbane Test."
Even if the incident is minor it is a bad look for the tourists, who are currently missing Ben Stokes because of the allrounder's brawl outside a Bristol nightclub that left another man hospitalised with facial injuries.
England coach Trevor Bayliss flagged the prospect of an Ashes curfew following the Stokes fracas but decided to give his players free rein, trusting them to enjoy themselves in a sensible manner.
English players also enjoyed beers in Adelaide and Townsville, where the visitors played their other tour game prior to landing in Brisbane for the first Test.
"There are no set curfews, they are just sensible rulings," Bayliss said earlier this month.
"Not drinking between matches is just sensible. We certainly don't want to keep players in their rooms because it is a long tour.
"You have to get out and experience what the country you're touring has to offer.
"It's about picking the right time to have a couple of drinks, but knowing to stay away from it if you're preparing for a match."
The clash between Bairstow and Bancroft comes four years after a bar-room scrap between David Warner and Joe Root preceded an Ashes series in England.
Warner was effectively suspended for two Tests after he clipped Root's chin in a nightclub.
Australia's vice-captain claimed he was angered by Root's impersonation of Hashim Amla.
CA warned Warner at the time he was on his last chance.
- AAP