Australia were dumped out of the Rugby World Cup in a 40-16 quarter-final defeat to England on Saturday.
Cheika brought his five-year stint in charge, which included a run to the World Cup final in 2015, to an end after the defeat.
New Zealand coaches Jamie Joseph and Dave Rennie have been linked with the post but, when touching down after flying back from Japan, Cheika told reporters Rugby Australia should look at home for their next appointment.
"I think definitely we should be pushing for an Australian coach," he said.
"It's not up to me but I think we should be backing and supporting Australian coaches wherever possible."
Cheika had said before the tournament that he would step down if Australia failed to lift the trophy and he insists there was never a chance he would change his decision.
24 - @EnglandRugby 's 24-point victory was their joint biggest Test win against Australia, in fact their three biggest victories against the Wallabies have come under Eddie Jones. Masterful. #RWC2019 #ENGvAUS pic.twitter.com/NugKbW99tB
— OptaJonny (@OptaJonny) October 19, 2019
"We came second last time and I figure [after] four years you've got to come first next time," he added.
"You can't call it and then change your mind afterwards because that's genuinely what we wanted to do - go there and win."
Skipper Michael Hooper paid tribute to Cheika's contribution on and off the training field.
"Cheik's been amazing, me personally I owe that man a lot," he said. "The passion that he represented us or stood up for us all the time and just generally wanted the best for Australian rugby.
"Not just for the team, not just for him selfishly being the coach of the team but wanting the best for Australian rugby.
"After he's long gone, to leave something that's positive, he's always believed in that and I think he will.
"He's made me a better person, not just a rugby player. So, I've got a lot to thank him for that."
Photo Getty Image Caption: Departing Australia coach Michael Cheika