The 55-year-old became the frontrunner for the Parkhead post following the breakdown in talks with top target Eddie Howe 10 days ago, and a deal now appears imminent with discussions at an advanced stage.
The Scottish Sun reports that an appointment is due 'within the next 72 hours' after Postegoclou reached a verbal agreement with Celtic over the terms of his contract.
It follows a compensation package being agreed with his current club, Yokohoma F Marinos, prior to the weekend.
Postecoglou would not be drawn on the speculation when he faced the media after his side’s 1-1 draw with Consadole Sapporo in the first leg of the J. League Cup during Saturday night.
Asked if he could say anything about leaving Marinos, he said: "Nah, not really.
"I think last week I was asked the question too. My focus is always on the work I have to do today.
"We had a very important game and we've got another game coming up on Wednesday so that's where my focus is."
The Celtic squad return for pre-season training in the third week of June ahead of their opening friendly fixture against Sheffield Wednesday on July 7, with the crucial Champions League qualifiers due to begin two weeks later.
However, red tape issues may delay the new man taking charge of the Glasgow club, as Postecoglou would require to self-isolate upon arrival in Scotland while the club is also seeking an exemption from UEFA over his lack of European coaching qualifications.
Meanwhile, striker Odsonne Edouard will leave the club for Leicester City this summer after a fee was 'agreed' for the striker, according to a report.
The Sunday Times claim that the Frenchman is set for a reunion with Brendan Rodgers, the man who brought him to Celtic Park from PSG in a £9m transfer in 2018, with an £18m figure quoted.
Arsenal and Inter Milan have also been credited with an interest in the 23-year-old striker front man, but the Foxes now appear to have won the race.