Postecoglou, who took over the role after Nuno Espirito Santo left the team, has had a tough start to his tenure.
He has managed only one draw and five losses in six games, failing to secure a win as Forest's manager.
Marinakis, who is close to Postecoglou, has reportedly told the manager that he expects improvement and will not hesitate to make changes if results don't get better.
Despite this, the Forest board is standing behind Postecoglou for now, as the team prepares to face Newcastle United at St James' Park on October 5.
A loss would make Postecoglou the first permanent coach since 1960 to start his Forest career with four consecutive league losses.
