The BCB said the 23-year-old pace star suffered a side injury during the second T20 match against Zimbabwe last week.
"Rana is expected to require around four weeks for tissue healing, followed by another two to four weeks of bowling workload progression, putting his overall recovery period at six to eight weeks," it said in a statement.
Left-arm pace bowler Shoriful Islam, who also sustained a hamstring injury during the Zimbabwe tour, could make a comeback in the second Test, subject to medical clearance.
The 25-year-old would "undergo a fitness assessment around August 12th," the BCB said.
Wicketkeeper Litton Das could also be available for the second Test if he recovers from a calf muscle injury, it said.
Bangladesh leave for Australia on July 31st, with the first Test beginning in Darwin on August 13th. The second Test will be played in Mackay from August 22nd.