Celtic have turned towards the former Australia coach after talks with Eddie Howe collapsed and they are expected to appoint him.
The Glasgow giants have agreed a compensation package with Yokohama F. Marinos and want UEFA to hand Postecoglou an exemption due to the lack of a UEFA Pro License.
Moore is aware that many people in Scotland do not know a lot about Postecoglou, but he has insisted that the coach has proved his credentials in Australia and Japan, and is a winner.
The former Rangers star claimed that he is a modern coach whose teams play the kind of football that managers such as Klopp, Bielsa and Thomas Tuchel would appreciate.
He believes his style of football will suit Celtic and he is likely to take them forward after a disappointing campaign.
Moore said on the Go Radio Football Show: “He is a coach that I know a lot of people in the UK struggle to maybe find out or know too much about.
“But if I am being honest then that is a bit lazy because when you do a bit of research, he has been a winner at Brisbane Roar, he went on a 36-game unbeaten run and won trophies there.
“He then went on to Japan at Yokohama where they were a mid-table team and in the second season, he won the J-League title.
“He is a winner and the type of football he likes to play is very entertaining and high energy like a lot of the modern-day coaches like the Klopps, Bielsas and the Tuchels.
“He likes to play that style of football, his teams like to dominate the ball and create a lot of opportunities.
“I personally think he would be a good fit for Celtic, I think he will improve them after a really disappointing season.”
The clock is ticking on Celtic’s players reporting back for pre-season, while the Bhoys also have European qualifiers to tackle.