Cummins was ruled out of Australia’s limited-overs series against New Zealand and India last month after scans revealed a lumbar bone stress in his back.
He has a history of back injuries, with flare-ups in 2012-13, 2013-14 and 2015-16 causing him to miss three Australian summers in four years.
He said he would be “aggressive” and “take risks” in his recovery in order to play in the Ashes.
Bailey said he did not know if Cummins had begun bowling as part of his rehabilitation but that it was “getting tight” for him to be fit for the first Test.
Should Cummins not be fit in time for the opener in Perth, former captain Steve Smith will skipper the side.
All-rounder Cameron Green was recently pulled out of Australia’s squad for their one-day international series against India as a precaution because of soreness in his side.
Bailey described it as a “minor” injury and said Green will play in the Sheffield Shield, so will be “in a good place to be ready to bowl”.
“We put so much time and energy into building out the plan for him to be available for the first Test as an all-rounder,” Bailey said.
“He was only going to play the first two ODIs anyway. The way he swings the cricket bat in white-ball cricket, even if it just sets him back for 24 more hours, we just made the decision to approach this more conservatively.”