Australia's coach Andrew McDonald confirmed on Tuesday in Durban that Green had failed to overcome the injury, which he suffered in the final ODI last week.
The blow likely leaves Australia with a choice to make between Cooper Connolly or Nic Maddinson at No.5, with Travis Head to remain opening and Alex Carey at No.6.
The injury does come at a cruel time for Green, who had appeared to be finding some rhythm with a century against Bangladesh in Darwin in August.
"He won't be available for the first Test match," McDonald said of Green.
"He is progressing pretty well, all things considered ... He is still here for a reason, so there is some light at the end of the tunnel.
"It will just be a matter of going through the rehab, he has a pretty demanding week as he reintroduced some activities.
"It's a bit of an unknown, but there is still optimism that he will take part in the series at some stage."
Green had been managing a shoulder issue that stopped him bowling in the white-ball games, with a planned bowling return in the final ODI aborted on the abdominal problem.
"It's highly unlikely (he'll bowl in this series)," McDonald said.
"With where he was with his shoulder injury, he missed a fair bit of work there.
"We would be solely focused on getting him to return as a batter, we wouldn't be putting that demand on him to return as a bowler.
"That would be too high risk is the information I have."
Maddinson has not played a Test for Australia since 2016, but was a surprise addition to the squad last week when Josh Inglis suffered a fractured finger.
Australia are also weighing up whether a dry Durban surface pushes them towards picking Connolly, with an expectation the Kingsmead wicket will turn later in the match.
Matt Kuhnemann is also on the tour as a second spinner, but it would appear unlikely that Australia will opt to play him ahead of one of the three frontline quicks.
Green's injury also increases the chances of Beau Webster holding his spot as a fourth seamer, after he played in the last three Tests batting at No.7.
There is also a possibility the tourists could go in with two frontline spinners, but that appears unlikely.
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