Postecoglou’s stock has been on the rise due to the way Celtic have performed under him in Scotland and the kind of attacking football they played in the Champions League.
Frank Lampard’s future at Everton has come under the scanner due to the poor run of form they had before the World Cup break and there are suggestions that the Toffees are planning contingencies.
And there are claims that Postecoglou is the manager Everton are likely to go for if they decide to sack Lampard at some point this season.
Moore admits that the Celtic boss did walk away from the Australia job after helping them to qualify for the World Cup.
But stressed he does not see him leaving the Scottish champions in the middle of a season and jeopardising their stability.
The Australian said on the Go Radio Football Show: “I’d think that’s exactly what Ange would do [and not leave Celtic in the middle of a season].
“He qualified with the national team, but then came away from the actual World Cup.
“Then again, the qualification process was complete and they still got an opportunity to bring in someone else for the World Cup.
“I don’t believe that he would make a decision that would impact the team throughout a season.
“Therefore, I don’t think halfway through the season he would leave the football club.”
It remains to be seen whether Everton decide to sack Lampard and make a move for Postecoglou in the middle of the season.