Warner suffered the injury when he got in a tangle trying to play a Steven Finn bouncer at Lord's almost a week ago, forcing him to miss the rest of the one-day international series against England.
The opener's injury was examined by a hand specialist in Sydney on Thursday and Cricket Australia said while surgery was not required he would not be fit in time for Bangladesh.
"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," Cricket Australia's Pat Howard said.
Australia are due to play two Tests in Bangladesh next month, the first such series between the nations since 2006.
"It's really disappointing. 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," Warner said.
Cricket Australia said Warner was expected to be fit in time for the home series against New Zealand starting in November.
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