Cooper had signed a three-year deal with his Super Rugby club Queensland Reds in June but that deal, reportedly worth around £260,000 a year, is contingent upon a contract with the ARU with whom the 24-year-old has endured a fractious relationship.
Channel Nine news said on Monday the New Zealand-born fly-half had rejected a low incentive-based ARU contract, and could now switch to the lucrative French Top 14 competition or to Rugby League.
New Zealand World Cup-winning centre Sonny Bill Williams quit Union and signed for the Sydney Roosters in Australia's National Rugby League last week and said earlier this year he would only switch codes if Cooper followed suit.
"I reckon (Cooper) would be a huge success," Williams, also New Zealand's heavyweight boxing champion, said in September.
"He has definitely got the skill-set to play the 13-man game.
"The biggest difference is in rugby you are playing on the advantage line, whereas in league the defence is 10 metres back. Imagine what he could do with that extra space."
Cooper, currently sidelined by a knee injury, has not played for Australia since describing the team environment as "toxic" - a comment which led the ARU to fine him £40,000 last month.
Cooper apologised after the fine, which included a further £13,000 suspended for two years, the latest black mark on his troubled spell with the Wallabies.
On the pitch and on his game, Cooper is one of the best fly-halves in world rugby but headlines have not all centred on his abundance of attacking-talent.
Cooper was fined £6,500 by the ARU in 2009 after breaking the window of a taxi, with a burglary charge against him dropped the following year after a justice mediation process.
He led the Queensland Reds to a first Super Rugby title in 2011 after forming a devastating partnership with scrum-half Will Genia and the two were also at the centre of the Wallabies success in the final Tri-Nations that same year.
Cooper accepted the role of public enemy number one when he returned to his homeland for the 2011 World Cup but a low-key tournament for the fly-half ended with knee reconstruction surgery after injury in the third-placed play-off win over Wales.
In his absence, Melbourne Rebels back Kurtley Beale has showcased his talents at the fulcrum of the Wallabies side with Berrick Barnes, Mike Harris and James O'Connor able alternatives that have shown coach Robbie Deans that Cooper is no longer irreplaceable.
While a switch to League remains the most likely destination for Cooper, the lucrative French league, where former Wallabies fly-half Matt Giteau plies his trade, is another viable option.