Starc featured in Australia's failed Champions Trophy campaign but scans revealed the left-arm paceman has not fully recovered from the foot fracture that cut short his India tour earlier this year.
"Mitchell was due to go for follow-up scans after the Champions Trophy, as part of his recovery plan," physiotherapist David Beakely said after CA named a 13-member squad.
"He will now undertake a period of rest from bowling with the aim to have him return for the one-day series in India in late September as he begins his build up to the Ashes."
Fit-again James Pattinson joined fellow pacemen Josh Hazlewood and Pat Cummins, while the selectors will name a fourth fast bowler later.
O'Keefe was a notable omission, having taken 12 wickets in the first test in India and finishing as Australia's joint leading wicket-taker in the series, which the hosts won 2-1.
"Whilst Steve O'Keefe bowled well in Pune, he did not maintain this level in the remaining matches of the series..." national selector Trevor Hohns said.
Left-arm spinner Ashton Agar has been asked instead to partner off-spinner Nathan Lyon for the tests in Dhaka (from Aug. 27) and Chittagong (from Sept. 4).
"Ashton has continued to impress us with his form and we believe his bowling is at a level where he deserves to be playing on the highest stage," Hohns said.
"He will work nicely in tandem with Nathan Lyon and also brings a great all-round package to the team."
All-rounder Hilton Cartwright has also been named in the Steve Smith-led squad.
Australia postponed their tour of Bangladesh in 2015 over security concerns but CA executive general manager Pat Howard was satisfied with the assurance from their hosts.
"The Bangladesh Cricket Board and the Bangladesh Government have provided satisfactory levels of assurance and security for this tour to proceed at this stage, and we’re very grateful to them for this," Howard said.
"We will continue to work with them to finalise plans but also monitor advice from Government agencies and our own security advisors about the security risk."
Australia squad:
Steve Smith (captain), David Warner (vice-captain), Ashton Agar, Hilton Cartwright, Pat Cummins, Peter Handscomb, Josh Hazlewood, Usman Khawaja, Nathan Lyon, Glenn Maxwell, James Pattinson, Matt Renshaw, Matthew Wade (wicketkeeper).
(Reporting by Amlan Chakraborty in New Delhi; editing by Peter Rutherford)
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