"It's a player we've been watching for years, who could have been the best in the world during the last World Cup, had he not been injured," owner Mourad Boudjellal told French media.
He said manager Bernard Laporte would decide who would be his first choice flyhalf, with four others in the squad already.
"We hesitated a lot before signing another one. But we saw in the early season that we missed something and that it was not enough to have four flyhalves," he added.
Former Wallabies playmaker Matt Giteau is already at the club, who have reached this year's European Champions Cup final and will be chasing their third title in a row when they play Clermont Auvergne at Twickenham on May 2.
Cooper's manager had said on Thursday the player was still 'deciphering' new rules about the eligibility of overseas players for the Wallabies.
Australia, which had previously considered only home-based players for selection, partially reversed their policy this week to allow players to continue their test careers if they have played 60 tests and seven years of Super Rugby.