Alexander played in 14 of Australia's 15 tests last year but the decision was made to rest him for the June series after the ACT Brumbies' 37-10 victory over the Melbourne Rebels on Saturday, when he started on the bench.
"The injury is one I understand quite well being a former tighthead prop and while not a sinister issue, it can significantly influence performance due to the impact it has on a player's overall strength," Wallabies coach Ewen McKenzie said in a news release.
"In Ben's instance, it has been admirable to see him push through the pain at a Super Rugby level.
"However, when you consider the challenge we'll face against an extremely physical and abrasive French scrum, we all agreed it would be better suited for Ben to use the upcoming period to rehab the injury and re-gain his strength."
Australia open the series against France in Brisbane next week with two more tests to follow in Melbourne and Sydney.
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