However, the Lions reckon the Monaghan man could still play against Australia later this month if the surgery, which involves the insertion of three pins, is successful.
Head coach Warren Gatland said: "We will just have to wait and see how the surgery goes. The surgeon says if it goes right, it could be just a couple of weeks."
If the Ireland winger can recover in that time frame, it would give him a chance of possibly returning for the closing two matches of the Test series.
Gatland will see how the operation goes before deciding if Bowe will continue in Australia or depart the tour, showing what high regard the 29-year-old is held in by the Lions coaching staff.
Another of the Irish contingent, out-half Jonathan Sexton, picked up an injury in the 22-12 defeat of Queensland. He underwent a scan on his hamstring and will play no part in next Tuesday's match against the NSW-QLD Combined Country.
But the Dubliner should be back in action soon with Gatland admitting: "It's just a bit tight, but he is okay."
Sexton's Leinster and Ireland colleague, Rob Kearney, is the last member of the original Lions squad yet to play on the tour Down Under.
The full-back is still recovering from a grade one hamstring tear, but his progress has pleased the Lions boss and he should be fit to play against the NSW Waratahs next weekend.
"It was good to see Rob running around yesterday and kicking a ball. Hopefully he is going to play a full part in training and be available for selection," added Gatland.