The mercurial 27-year-old, who played his 54th test for his country in Saturday's victory over South Africa in Brisbane, had signed a "pre-commitment" to join the European champions in April.
However, there has been ongoing speculation that his agent, Khoder Nasser, was trying to find a way to keep Cooper in Australia.
Negotiations with the Queensland Reds were peremptorily brought to an end late last month with Queensland Rugby Union chief Jim Carmichael citing Nasser's "unrealistic expectations".
The Australian Rugby Union (ARU) were also keen to keep Cooper at home, however, and Sunday's Sydney Morning Herald report said he would sign a four-year deal, concentrating on sevens next year and joining a Super Rugby team for 2017.
Wallabies coach Michael Cheika said on Sunday he had not been keeping on top of the deal while preparing his side for the Rugby Championship but had always hoped Cooper would change his mind.
"I've said consistently that I'm keen for him to stay and be in the picture," Cheika said.
"I'm hoping he's going to be in Australian rugby for the longer term. With us helping him get to where he wants to go and become a better player, I think we've yet to see the best of Quade Cooper."
'INCREDIBLE BALL'
With the cash-strapped ARU already having said they would not help pay to get Cooper out of his Toulon contract and no Super Rugby club part of the deal, the player himself may have to foot the bill, which is reported to be around $150,000.
On Saturday, Cooper showed flashes of his undoubted skill in a patchy performance against the Springboks.
His most audacious contribution, a behind-the-back pass to Tevita Kuridrani out of defence in the dying moments of the first half, almost resulted in a try to South Africa when his team mate failed to catch it.
Cheika, however, said he would always back his players to try such moves.
"Tevita takes that ball, he’s away. There’s no way I’m ever going to tell a player not to do that because he can do it and he made the pass," he added.
"Tevita was probably shocked that it came his way because it was an incredible ball. If you get that in the hands, we score at the other end of the field on halftime.
"The lad practices that pass at training. He thought it was on and it was."
(Editing by Peter Rutherford)
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