Cooper's selection comes despite him playing only twice for the Queensland Reds since seriously injuring his knee at last year's World Cup in New Zealand.
However, captain James Horwill is set to miss out after hurting his hamstring in Queensland's 13-12 win over the ACT Brumbies in Canberra on Saturday.
Horwill has reported to the national team ahead of scans due on Monday, but is not listed among the 39.
Melbourne Rebels fly-half Kurtley Beale is also a doubt after suffering a shoulder injury for his side on Saturday.
There are a total of 15 uncapped players in the squad.
Cooper was one of the first players to report in Sydney as those selected from the Reds and Waratahs arrived early, with the remaining players to join in midweek before a team meeting on Wednesday.
They will then go back to their respective clubs for Super Rugby action before the final squad to play Scotland on June 5 in Newcastle is picked just 48 hours prior.
And Wallabies coach Robbie Deans thanked the Australian clubs for their assistance in releasing players, before admitting that such a short turnaround from selection to the game against the Scots was not ideal.
"The co-operation of the franchises has been considerable and we appreciate that," Deans said.
"We've obviously sought to have as little disruption to their campaigns as is possible, but the solution provided by utilising the training day off does at least allow us to get the players started - both with organisational logistics and training background - as we will have to hit the ground running once the final group gathers.
"We can't finalise a side for that match until all of the games are played, and we are in a position to assess all of the players.
"Obviously we have preliminary ideas but the selection template is a living document. It can, and probably will change, as a result of events through the final round of Super Rugby before the international break."
Following the one-off Test against Scotland, Australia will take on Wales three times in June before the Rugby Championship - involving New Zealand, South Africa and Argentina - begins in August.
Squad: Ben Alexander, Adam Ashley-Cooper, Berrick Barnes, Kurtley Beale, Quade Cooper, Dave Dennis, Kane Douglas, Anthony Faingaa, Saia Faingaa, Bernard Foley, Will Genia, Michael Harris, Scott Higginbotham, Michael Hooper, Rob Horne, Digby Ioane, Sekope Kepu, Salesi Ma'afu, Pat McCabe, Ben McCalman, Jesse Mogg, Stephen Moore, Luke Morahan, Cadeyrn Neville, Dan Palmer, Wycliff Palu, Nick Phipps, David Pocock, Tatafu Polota Nau, Hugh Pyle, Benn Robinson, Nathan Sharpe, Dominic Shipperley, Rob Simmons, James Slipper, Sitaleki Timani, Joseph Tomane, Cooper Vuna, Nic White
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