Former Bournemouth manager Eddie Howe had long been considered the front runner to replace Neil Lennon.
However, the PA news agency understands Celtic have now been forced to turn their focus elsewhere after the Englishman was unable to secure the backroom team he wanted to take with him to Parkhead.
Reports have emerged linking Postecoglou, 55, with the Celtic job. He is currently with Japanese club Yokohama F Marinos, having guided them to a first J League title in 15 years two seasons ago.
Postecoglou was in charge of the Australia national team between 2013 and 2017, leading the Socceroos at the 2014 World Cup before guiding them to Asian Cup success in 2015.
Celtic saw Old Firm rivals Rangers take the Premiership title, and whoever eventually takes charge will have to hit the ground running with a vital Champions League qualifier coming up on August 20.
On Friday night, Celtic confirmed the move for Howe was now off.
“Following very positive and detailed discussions with Eddie Howe, with the belief that he would be an excellent candidate for the position of Celtic manager, we allowed time for the process, given he’d previously made it clear he was not looking to return to management until this summer at the earliest,” the club statement read.
“We can now confirm that Eddie will not be joining the club, for reasons out of both his and Celtic’s control. We wish Eddie success for the future.
“As part of the ongoing process we have engaged with a number of candidates.
“We fully appreciate our fans will be seeking clarity on this matter and we aim, very shortly, to announce the appointment of a manager with the quality and desire to take Celtic forward and bring further success to our supporters.”