Forest owner Evangelos Marinakis and football chief Edu are seeking a new manager after the dismissal of Nuno this morning.
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Postecoglou is admired Marinakis and the Australian has remained in Europe since leaving Tottenham at the end of last season.
Jose Mourinho, sacked by Fenerbahce last week, is also being linked with the position.
Meanwhile, two under contract coaches who Marinakis admires are Athletic Bilbao's Ernesto Valverde and Fulham manager Marco Silva.
Going with a Portuguese speaker is being discussed inside Forest given the club's focus on the Portuguese and Brazilian transfer markets in recent years.