Scans identified an injury affecting a cervical facet joint, which plays an important role in maintaining stability and allowing movement in the neck. Despite the diagnosis, the 37-year-old is anticipated to be available when the four-match series begins in Perth on 9 December.
"Steve will continue to receive treatment and rehabilitation, with a full return to availability anticipated for the upcoming home test series against New Zealand," Cricket Australia said in a statement.
Smith was forced to miss Australia's deciding Test against South Africa in Cape Town after struggling with the problem for several days.
The discomfort first emerged during the second Test in Gqeberha, where he was moved down the batting order during Australia's unsuccessful attempt to save the match.
His condition deteriorated ahead of the third encounter, with the batter visibly uncomfortable during a nets session on the eve of the contest.
By the opening morning, his neck was heavily bandaged, and he was subsequently withdrawn from the playing XI as part of four changes to the Australian side.
Cameron Green returned from injury to replace Smith at number four, taking over the position normally occupied by the experienced batter.
Zubayr Hamza and Dane Paterson have been recalled to the Proteas starting line-up as the hosts made two changes for the first Test against Bangladesh, starting at the Wanderers on Sunday.
Marco Jansen has been named ICC Player of the Month for October after playing a crucial role in the Proteas' Test series victory against Australia.
David Warner claims Cricket Australia's (CA's) investigation into the 2018 ball-tampering scandal "stinks of a cover up" after alleging that some bowlers were not interviewed during the process.
The Proteas will select one of Tony de Zorzi or Zubayr Hamza to bat at number 3 in the first Test against Bangladesh, starting at the Wanderers on Sunday, according to head coach Shukri Conrad.
