The Australian coach's job security remains uncertain, despite leading the team to the Europa League final last month.
Tottenham ended the season with a disappointing home loss to Brighton and a 17th-place finish in the Premier League table.
Chairman Levy and Postecoglou have been on a break, but they are set to meet in London soon.
Last month, Postecoglou expressed his desire to continue as manager, saying he wants to stay and complete the job he started.
He had one goal in mind when he took the role: to win a trophy and build on that success.