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Toulon announced in April that the mercurial back had signed a two-year deal to join the Top 14 rugby club after this year's World Cup in England.
Despite that announcement, the ARU and his Super Rugby side the Reds have continued talks and Cheika believes there is now a deal on the table that could tempt Cooper to remain in Australia.
"The way he is investing in what we are doing... and his input into the programme and what is going on, you'd like to think he is staying for quite a while because that's definitely the feel he has given off," Cheika told reporters on Monday.
"His commitment and investment in that project, what he's giving, and his involvement there, and his eagerness to be involved.
"But it is still a decision that is up to him and his management. They are taking their time to make the right one and I don't have a problem with that.
"I'm very keen for him to stay."
Faced with a flood of internationals heading off to Europe after the World Cup, including Cooper's Reds team-mates Will Genia and James Horwill, the ARU changed its rules on eligibility in April.
Cooper currently falls seven caps short of the 60-cap tally for Australians who want to continue playing for their country while at clubs in other countries.
The 27-year-old missed most of last year's international season because of injury and slipped behind Bernard Foley in the pecking order for the Australia number 10 shirt.
He played only five matches for the Reds this year because of further injuries but showed a few flashes of his old brilliance in the last three games of the season after returning from a shoulder problem.
