Cheika returned to Sydney on Tuesday still disappointed by the Wallabies' quarter-final exit from the Rugby World Cup, but with no regrets about standing down after five years at the helm.

Cheika was on record months ago as saying he would not seek a reappointment if he couldn't take the Wallabies one step further than their runner-up showing at the 2015 edition in England.

"We came second last time, right, and I figure (after) four years you've got to come first next time," he said at Sydney airport.

"The way I see it, if you want to improve, you've got to call it and you've got to stay with your call.

"You can't call it and then change your mind afterwards because that's genuinely what we wanted to do: was to go there and win."

New Zealand pair Dave Rennie and Jamie Joseph are considered the two frontrunners to replace Cheika, but the outgoing mentor would prefer a home-grown coach to fill the hot seat.

"I think definitely we should be pushing for an Australian coach," Cheika said.

"It's not up to me but I think we should be backing and supporting Australian coaches wherever possible."

As admirable as that notion may be, Australian coaches with the credentials to take charge of the Wallabies are thin on the ground.

Eddie Jones, arguably the world's best mentor, remains contracted to England for two more years, while the Brumbies' Dan McKellar probably needs more time to develop at Super Rugby level.

Ireland-based former World Cup winner Stephen Larkham, who Cheika effectively had removed as Wallabies attack coach after a diabolical 2018 season, looms as the most likely Australian candidate.

"I've had Bernie (Larkham) at the Brumbies. Enjoyed my time there," said retiring Wallabies flanker David Pocock on Tuesday when asked about Larkham's qualifications.

"My understanding is he's pretty well locked in overseas (with Munster).

"All those things are above our pay grade. That's for the people at the top to be deciding and setting the agenda."

Cheika revealed on Sunday he barely had a relationship with Rugby Australia chief Raelene Castle, but declined to add fuel to the fire when asked if she too should fall on her sword.

Former Wallabies coach Alan Jones says Castle and the entire board must go as calls grow louder for a complete overhaul at the top of Australian rugby after a disastrous couple of years for the code.

"It's an irrelevant issue, honestly," said Cheika, insisting the fractured relationship with Castle and chairman Cameron Clyne couldn't be blamed for Australia's worst-ever World Cup campaign.

"It's for other people to decide all that. My job's with the team.

"That's where it's got (to be). No excuses. No nothing. That's the way I've always operated and that's the way I'll continue to operate going forward."

Australian Associated Press