It is understood that Bancroft, who at that stage had not been selected for the Australian team, did not retaliate.
Bairstow is now facing questions over his behaviour just months after Ben Stokes and Alex Hales were involved in an early-hours brawl after a night out in Bristol.
The comments, which centred around the pair telling Bairstow "we know what you did to our mate in Perth" might have contributed towards a careless stroke from one of England's best batsmen as they crashed to near defeat.
Fox Sports Australia is reporting Bairstow headbutted Bancroft, while other versions suggest it was more low key and a case of tomfoolery gone awry.
The 35-year-old was backed by the English media who argued the reports were simply trying to stir the pot with no facts behind them.
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The latest explosive development came to light at the end of the fourth day of the first Test after Bairstow had been sledged aggressively by Australia, in particular David Warner and Peter Handscomb, when he batted at the Gabba.
"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".
The Bairstow incident is potentially another major setback for an England side that is nearly certain to head to Adelaide one down in the five-match series. The incident allegedly took place earlier this month at a Perth hotel bar while England were in town for a tour match, reports said.
Another source told Sportsmail 'Jonny is a rugby man and part of their culture is to bump heads if someone buys them a drink.
The incident could have implications for Bairstow, who is now in Brisbane for the Ashes tour.
But he was one of three England players fined by the European Central Bank in the wake of a night out in Bristol that saw Ben Stokes arrested for alleged assault.
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