While winning the Europa League has seen Spurs qualify for next season's Champions League, they still finished the Premier League season one place above the dropzone in 17th place.
As such, Levy is weighing up whether to go with Postecoglou next season or replace the Australian.
The Mirror says the players are urging Levy to stick with the manager, who has argued their season was affected by an "unprecedented" injury crisis.
I always believed Ange would deliver
For his part, midfielder James Maddison is eager to continue working for Postecoglou: “The start of the season when he says I don’t usually win things, I always win things. We knew that would create uproar in the media. Never once did I stop believing he meant that.
“He’s been brilliant. The season three line was iconic as well, that was the ultimate mic drop moment. His biggest strength is his self belief in what he does and that feeds off into us and it’s been installed into us into Europe this year.
"Managers and clubs can go long periods without winning trophies - you don’t need a bigger example than Tottenham - but he’s a manager who has always had success.”