Rees is set to be out for two weeks, which would also mean him missing the Millennium Stadium appointment with Scotland on Sunday week.
The former Wales captain is in a battle for the number two shirt with Huw Bennett, with the latter set to win his 50th cap in Dublin.
Rees' absence now means a probable place on the bench for his Scarlets team-mate Ken Owens, who made his Test debut during the World Cup in New Zealand.
The injury to Rees is the latest blow to Wales, with fitness doubts remaining over centre Jamie Roberts, fly-half Rhys Priestland and flanker Dan Lydiate.
Wales coach Warren Gatland has already lost prop Gethin Jenkins and his World Cup locks Luke Charteris and Alun-Wyn Jones due to injuries as they prepare for a World Cup quarter-final rematch against Ireland at the Aviva Stadium.
Gatland will announce his team on Friday.
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The Lions are ready to get their Australia tour underway. They arrive in Perth on Monday.
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