This comes after Nuno Espirito Santo was dismissed by the club following strained relations with the hierarchy.
The former Celtic manager has been strongly linked in recent weeks while Santo's future looked uncertain and he was instantly placed as favourite after the Portuguese moved on.
Brendan Rodgers has also been linked but it looks like his Parkhead predecessor will land the job.
The Telegraph reported that Postecoglou is expected to be appointed this week, with a recent update suggesting this could happen within the next 72 hours.
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He is said to be the preferred choice of Forest owner Evangelos Marinakis, over Rodgers and Jose Mourinho.
Marinakis is believed to be keen on the 60-year-old's ability to deliver an "expansive, attacking style of football."
An appointment is expected to be made before Forest's lunchtime match against Arsenal this Saturday.
Santo had openly expressed concerns about his relationship with Marinakis and global head of football, Edu.
Edu had previously admitted there was "no smoke without fire."
This morning, Forest released a statement confirming Santo's departure.
The statement read: "Nottingham Forest Football club confirms that, following recent circumstances, Nuno Espírito Santo has today been relieved of his duties as head coach.
"The club thanks Nuno for his contribution during a very successful era at The City Ground, in particular his role in the 2024/25 season, which will forever be remembered fondly in the history of the club.
"As someone who played a pivotal role in our success last season, he will always hold a special place in our journey."
Postecoglou was sacked by Tottenham Hotspur four months after winning the Europa League due to poor league performance.