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The tweets, now erased from his feed @quadecooper, suggested there were problems behind the scenes at the ARU and that he was not allowed to play the brand of rugby he preferred with the Wallabies.
The New Zealand-born flyhalf followed that with an interview with local media on Sunday in which he described the environment around the ARU as "toxic".
In brief statement on Thursday, the ARU said it had written to Cooper "in relation to social media comments" but had "no intention of conducting those discussions with Quade in the public arena".
The 24-year-old Wallabies and Reds back has been sidelined for Australia's last two Tests of the rugby championships in South Africa and Argentina because of a knee problem.
The ARU has also declined to discuss the progress of a new Wallabies contract for Cooper, whose current deal expires at the end of this year.
Cooper has signed a new three-year contract with the Reds but it will not be ratified until a Wallabies deal is completed.
His recent comments along with another saying he wanted to play with Sonny Bill Williams - the All Black who has been linked with rugby league's Sydney Roosters - has further fuelled rumours that Cooper might be preparing to quit the code.
At his best a mercurial talent whose unpredictability can unpick the game's increasingly tight defences, Cooper helped the Reds to their first Super Rugby title and Australia to a first Tri-nations title in a decade last year.
Reds rugby chief Jim Carmichael was moved to release a public statement on Wednesday in an attempt to ease the fears of the province's fans.
"Let me reassure you that Quade has no interest in leaving the Reds," it read.
"He loves playing for the Reds and representing his country and has
reaffirmed his commitment to our organisation on several occasions, and
again as late as Wednesday afternoon."
