Postecoglou was appointed at Spurs this summer after spending the past two seasons with the Glasgow giants, where he won the Scottish title both years and the domestic treble before leaving.
Tottenham is the most high-profile job of the 57-year-old’s career, though, and he has the task of rebuilding the club after a disappointing season.
Moyes, however, has backed Postecoglou to cope with the spotlight after dealing with the heat when coaching Celtic.
“He will never get more pressure than he did managing Celtic Football Club,” said Moyes, who began his playing career at Parkhead.
“Celtic Football Club are probably the third or fourth best supported club in the world, and if you manage Celtic Football Club you are used to pressure. So he will have no problem.”
West Ham take on Tottenham on Tuesday here in Perth as they gear up for the new season.
Moyes was linked with leaving the Hammers at the end of last season, but he has stayed on after winning the Europa Conference League.
And asked about reports he held talks about his future, Moyes said: “I am used to having lots of talks with the owner. I speak to him most days.
“So it was nothing special – any talks at the end of the season. I wouldn’t have put the sort of titles you are suggesting there alongside it.”
