Postecoglou's brief reign saw the team play eight matches without a win, recording six defeats and two draws.
Fulham's Marco Silva is a primary target, with Forest owner Evangelos Marinakis having previously worked with him at Olympiakos.
Fulham would reportedly demand £10m for Silva, who is said to be reluctant to move mid-season.
Other candidates include Sean Dyche, a boyhood Forest fan seeking a long-term contract, and Euro 2020 winner Roberto Mancini, who would command a high wage.