It's claimed Spurs chiefs have even drafted a list of possible successors to Postecoglou should they decide to part company with the former Celtic manager.
The Premier League side currently sit 14th in the league table, with 10 wins, four draws and 15 defeats so far this season.
The Athletic reports Spurs officials are determined to finish the season strongly but there are question marks over Postecoglou's position beyond the summer.
The manager, 59, has another year left on his deal come the summer, with Spurs holding an option for a further season.
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However, it's thought Postecoglou is under major scrutiny over his position.
According to reports, Bournemouth manager Andoni Iraola has been made the top target for Spurs should they relieve Postecoglou of his duties.
It could cost £10m to take Iraola from his current role, where his side sit ten points ahead of Spurs in 10th in the league table.
Fulham manager Marco Silva and Brentford boss Thomas Frank have also been branded possible options for Spurs.