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.
Bancroft didn't play in that game but was at home prior to flying out for a Sheffield Shield match in Sydney.
There are differing versions of what unfolded.
Fox Sports News' Neroli Meadows broke the story late on Sunday night, revealing that Bairstow was being investigated by the England Cricket Board over an incident which occurred in Perth at the start of their tour of Australia.
As the European Central Bank noted, there are no police reports, but neither the European Central Bank nor Cricket Australia were commented further when approached by reporters over the matter on Sunday. Bairstow since then has apologised to Bancroft.
"At close of play in Brisbane today, we were made aware of allegations of an incident in Perth four weeks ago", an European Central Bank spokesperson said.
David Warner and Cameron Bancroft compounded an already exasperating day for England by sharing an unbeaten 114-run opening stand to move Australia within 56 runs of victory with a full day to play in the Ashes series opener. The incident allegedly took place earlier this month at a Perth hotel bar while England were in town for a tour match, reports said.
England coach Trevor Bayliss flagged the prospect of an Ashes curfew following the Stokes fracas but chose to give his players free rein, trusting them to enjoy themselves in a sensible manner.
The matter has not been pursued by Bancroft, but Bairstow was given a big send-off by a couple of Australian players when he departed meekly to virtually end England's hopes of posting a competitive target.
"Not drinking between matches is just sensible". You have to get out and experience what the country you're touring has to offer.
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.
In that instance, Warner was suspended for punching now-England captain Joe Root in an alcohol-fuelled incident.
Australia's vice-captain claimed he was angered by Root's impersonation of Hashim Amla.
Warner (60 not out) set about the run chase cautiously in the last session Sunday but picked up his scoring rate late to raise his 25th test 50 with a boundary.