Arthur was sacked shortly before the start of the ongoing series in England and is now seeking compensation, reported to be in the region of 4 million Aus dollars (£2.44m).
The South African, who was contracted until June 2015, was replaced by Darren Lehmann after a troubled tenure which saw numerous off-field disciplinary issues.
Four players were internally suspended for the third Test on the tour of India after Arthur and captain Michael Clarke questioned their attitude while opener David Warner was also banned for punching England's Joe Root in a Birmingham bar during the Champions Trophy.
According to reports in Australia, the court documents lodged in his compensation claim also include claims Clarke and experienced all-rounder Shane Watson are at loggerheads.
"We're disappointed that it's come to this position," Cricket Australia lawyer Dean Kino told Seven News.
"But Cricket Australia's confident in its position on this matter. And I'm sure it'll get resolved in the appropriate fashion."
The news comes a day after Warner revealed he felt he had played a role in Arthur's sacking.
"It was probably another thing that was gutting, that I may have played a part in that," Warner said before travelling to Zimbabwe and South Africa with the Australia A side.
Under Lehmann, Australia narrowly lost the first Ashes Test by 14 runs at Trent Bridge.
The second Test begins at Lord's on Thursday.
Former Australia leg-spinner Shane Warne believes Arthur's claim is "out of order" and said the South African should just have been happy he got the chance to coach Australia.
"It sounds like sour grapes to me and it's pretty diappointing," he told Sky Sports News.
"He should have been grateful for the opportunity to coach the Australian team.
"It's disappointing that he wants money like that, especially at this time. Go behind closed doors and do that if that's your gripe - and I think it is unfair to want that off Cricket Australia.
"I don't think it's up to him to want $4million. I think that's out of order."
Warne said CA should be praised for sacking Arthur and joked that his legal action was akin to the infamous 'Homework Gate' saga the former coach oversaw during the India tour.
"I'm one of Cricket Australia's harshest critics at times but I think they've been very good after the Champions Trophy," he said.
"You've got to give a tick to the selectors for picking the squad they have.
"I think you have to give them big ticks for being proactive with Darren Lehmann. He's a fantastic leader and I think that's showing already; even though they've lost they've showed a better spirit in the team.
"So Mickey Arthur should maybe go back to 'Homework Gate'.
"Maybe he should be writing some lines and some paperwork and send them into Cricket Australia.
"If I was him I'd just sit back on my couch and say 'I was lucky to have the opportunity'."