Mark Wood was named with fellow quicks Stuart Broad and Ollie Robinson but may end up making way for spinner Jack Leach depending on the Gabba wicket.
If Wood is preferred, Joe Root-captained England could go in with five seam-bowling options, with Ben Stokes and Chris Woakes included as all-rounders.
Anderson is fit but is being rested for the match in Adelaide, the team said.
"(It's) just a precautionary measure with an incredibly long series ahead," wicketkeeper Jos Buttler said.
England clarified that Anderson had no injury concerns after media reported he had been ruled out of the Gabba Test due to a calf strain.
With Buttler confirmed as wicketkeeper, middle order batsman Ollie Pope is set to be the beneficiary of Bairstow's omission and will have a chance to build upon his 81 in the fourth test against India at The Oval.
Buttler said Stokes, who missed the last Ashes in Australia in 2017/18, would be ready for a normal bowling workload after a long lay-off and a finger injury.
"Ben is fit to play. He's been performing very nicely in the nets," Buttler told reporters.
"It's a massive plus for us as a side."
Rory Burns and Haseeb Hameed are set to continue their opening partnership after combining for the last two tests of the home series against India.
England, who have not won at the Gabba since the 1986/87 series, are bidding to win the Ashes in Australia for the first time since claiming a 3-1 victory in the 2010/11 edition.
England squad: Joe Root (capt), Stuart Broad, Rory Burns, Jos Buttler, Haseeb Hameed, Jack Leach, Dawid Malan, Ollie Pope, Ollie Robinson, Ben Stokes, Chris Woakes, Mark Wood.