The all-rounder is now playing for Canterbury Kings with he and Hales having not played for England since 24 September after they were declared unavailable for selection until further notice following their night out in Bristol which saw Stokes arrested.
Bairstow, Joe Root, Moeen Ali, Chris Woakes and Jake Ball are the only members of England's test squad who will stay on for the ODI series. Ben Stokes and Alex Hales have been out of global action ever since they were allegedly found their involvement in the pub brawl in Bristol this September.
"Selectors were given clear guidance to name their strongest squad and have made their selections accordingly", an European Central Bank spokesman said.
Bayliss admitted Stokes's potential appearance in Australia could be a distraction for the team.
England coach Trevor Bayliss is expecting the selection to cop plenty of attention.
"Whether it's any more of a circus than it's already been, it's hard to say".
"We've got to take a decision now, and. the two guys who have taken over, I just think that's the best way to go".
Stokes' performance was at least an improvement on his Canterbury debut last weekend when he scored only two runs in a six-ball innings and went wicket-less in nine overs in a loss to Otago.
The England all-rounder made a brisk 34 against Auckland today.
The participation by Stokes and Hales in the series will however stay a subject to any relevant legal or disciplinary developments in relation to the incident in Bristol in September.
"I've got absolutely no idea".
A lot of the pre-Ashes talk was dominated by Ben Stokes and whether or not he'd be eligible for England Down Under.
The ECB said in their statement that once they received formal confirmation on whether Stokes had been charged or not, they would meet within 48 hours to decide the next step.
The news has come as a big relief for the English cricket fans.
Kent batsman Sam Billings and Durham seamer Mark Wood were also included in the squad that defeated West Indies (4-0) earlier this year.