The 32-year-old ruptured the medial ligament in his right elbow while throwing during an Australian T20 training session last month.
“AJ underwent successful reconstructive elbow surgery in Sydney a fortnight ago and has been recovering well,” WACA Sports Science Sports Medicine Manager Nick Jones said.
“We expect AJ will be unavailable as a bowler for 3-4 months.
“His availability in the BBL will be reviewed in the second half of the tournament.”
Head Coach Adam Voges said it was disappointing to lose such an experienced T20 campaigner.
“It’s never nice to lose any player to injury, but someone as experienced and well liked as AJ is a big blow for us,” Voges said.
“I’m confident we have the depth to cover AJ and hopefully fans will see him in action at The Furnace towards the end of the Big Bash.”