Cooper was taken off in the eighth minute of the Queensland Reds' 30-27 loss to the Melbourne Rebels in Brisbane after two players landed on top of him in a tackle.
He was later diagnosed with a grade three AC joint separation and could be out for up to six weeks, local media reported.
Wallabies coach Ewen McKenzie is due to name his 32-man squad on Thursday for the three-game series against France starting on June 7 in Brisbane, with the ACT Brumbies' Matt Toomua and New South Wales Waratahs' Bernard Foley now likely to come into test contention.
Waratahs utility back Kurtley Beale, who has played test matches at flyhalf but been in impressive form at inside centre for the Sydney-based side, also looms as a potential fill-in for Cooper.
Beale, who had a terrible 2013 in which he sought treatment for alcohol-related issues, was keen to play in the centres at international level to continue his developing relationship with fullback Israel Folau.
"If we get picked, I'd love to play with him for the Wallabies," he told Reuters earlier this week.
"I've had a great time playing with him for the Waratahs and we've got a combination going."
- Sports & Recreation
- New South Wales Waratahs
- Melbourne Rebels
- Queensland Reds
- Matt Toomua