Bairstow had broken his leg and dislocated his ankle when he slipped on a golf course in September last year.
England are already led by an Australian in head coach Matthew Mott who replaced Chris Silverwood earlier this year.
England are beginning a new leadership era in Test cricket under new captain Ben Stokes and new coach Brendon McCullum. England will play New Zealand in 3 Tests.
O'Keefe, who played nine Tests and seven T20Is, felt if the two-coach model could work for England, there was no reason why it won't work for Australia.
Ben Sawyer is also the head coach of Birmingham Pheonix in The Hundred and has previously coached Sydney Sixers in Women’s Big Bash League.
England Test Skipper and all-rounder Ben Stokes announced his retirement from the 50-over format earlier this year.
