Cheika confirmed he will not seek re-appointment after Australia were routed 40-16 by England in the World Cup quarter-finals on Saturday.
The 52-year-old, who guided the Wallabies to the 2015 World Cup final as he was named World Rugby Coach of the Year, bristled at questions over his future in the immediate aftermath of Australia's elimination.
However, former Waratahs boss Cheika quit on Sunday – ending his five-year stint in charge of Australia.
"It is no secret I have no relationship with the CEO [Raelene Castle] and not much with the chairman [Cameron Clyne]," Cheika was quoted as saying by the Sydney Morning Herald.
Cheika replaced Ewen McKenzie in 2014 and he made an immediate impact as the Wallabies reached the 2015 World Cup final – beaten by New Zealand.
That run to the decider saw Cheika become the first Australia coach to claim World Rugby's top coaching award since Rod Macqueen in 2001.
But the Wallabies' performances slowly regressed and pressure mounted on heading into this year's World Cup.
In a statement released by Rugby Australia, Cheika said: "I got asked the question in the press conference about what's going to happen going forward and at the time I wasn't keen to answer, but I always knew the answer in my head.
"I just wanted to speak to my wife and tell a few people up there [on the Rugby Australia board] about it.
"I put my chips in earlier in the year - I told people no win, no play.
"So, I'm the type of man who always goes to back what he says and I knew from the final whistle, but I just wanted to give it that little bit time to cool down, talk to my people and then make it clear."
New Zealander Dave Rennie – who is in charge of Glasgow Warriors having previously led the Chiefs to two Super Rugby titles – is the favourite to replace Cheika.