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.