Warner, whose opening partner Chris Rogers retired after the recent Ashes defeat, sustained the injury in a one-dayer against England last week and a hand specialist in Sydney has confirmed he will not be fit to make the trip.
"It's really disappointing," said Warner, who was recently appointed vice captain under new skipper Steve Smith.
"I never want to miss a match for Australia but I'll be doing everything I possibly can to get back as quickly as possible."
The 28-year-old is, however, expected to be ready to face New Zealand in the three-match series on home soil in November, which includes the first ever day/night test at the Adelaide Oval.
"David has been told that the fracture will take at least four weeks to stabilise and a further two weeks before he can start batting and catching again," said Cricket Australia's Pat Howard.
"He will be reviewed by the specialist at the end of the month to determine when he can resume playing. Ideally we'd like him to play the Sheffield Shield round beginning on 27 October as preparation for the New Zealand series."
With the test retirement of all-rounder Shane Watson and the death of Phillip Hughes at the end of last year, Australia have a dearth of experienced openers at their disposal.
Joe Burns, who played two tests against India in the middle order last season, and the uncapped Cameron Bancroft are contenders, while 18-test veteran Ed Cowan has continued to score runs after being dropped for Rogers in 2013.
The trip to Bangladesh was always going to be a transitional series for Australia after the retirement of captain Michael Clarke, stalwart wicketkeeper Brad Haddin, Watson and Rogers.
The tourists play a warm-up match in Fatullah on Oct. 3 before tests against Bangladesh in Chittagong (Oct. 9-13) and Mirpur (Oct. 17-21).
(Reporting by Nick Mulvenney; Editing by John O'Brien)
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