talkSPORT understands Celtic are bracing themselves for the Australian coach's departure as Spurs eye their next permanent manager.
Postecoglou has enjoyed a lot of success at Celtic since he took the job in 2021.
He could also lead the Glasgow giants to a treble this weekend as they face Inverness Caledonian Thistle in the Scottish Cup final.
Despite his credentials, many Spurs fans are less than impressed with the prospect of Postecoglou being their next manager.
As a result, some took to Twitter to launch #NoToPostecoglou in protest against the appointment.
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One fan said: "As a Tottenham season ticket holder for 30 years, we can't have this guy in charge. He'll relegate us!"
"Anyone welcoming this guy is either Australian or Levy's best friend. Only Levy inners would be happy with this appointment," said another.
A third said: "We are seriously going to blow out Pochettino, let him go to Chelsea for the Aussie Nuno. Levy is sabotaging Tottenham Hotspur. He will never be forgiven."
Former Spurs and Celtic defender Ramon Vega didn't contribute to the hashtag, but revealed his belief that Postecoglou should stay in Scotland.
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He tweeted: "If Ange Postecoglou wins the treble for Celtic, the next challenge shouldn't be managing Spurs but fighting for playing in the Champions League.
"Or [he should be] going to a reputable big European club, which Spurs unfortunately is for sure not one of them at the moment."
Amid the news of Postecoglou's potential move to Spurs, talkSPORT's Simon Jordan revealed his belief that the current Celtic boss would improve to north London side.
"I think he's a very capable football manager," said the White and Jordan host. "I think often with Antipodeans, you get this culture of an ability to engage and deliver and concentrate the minds of their charges. So, I'm not surprised he's been successful at Celtic.
"Now, stepping from Celtic into Tottenham, do I think he would improve Tottenham? Yes. Do I think he would square the circle of getting them to the holy grail of being able to deliver something? I'm not sure.
"It's a big departure from managing in one hemisphere of the world, landing in Celtic and running yourself into a two-tier football league and not being overly competitive in Europe, where the standard of the European competition would be up there with the standards of the Premier League, and they [Celtic] didn't excel in that space.
"But, I think he's well worth proper scrutiny and consideration. Whether Spurs are doing that, don't know, can't speak to that.
"But I wouldn't sit there and go, 'Woah, hang on a second, no no no, where are you going, you're shopping in Poundland, you need to be shopping somewhere else'.
"No. I think he's a proper football manager that is more than capable of taking the next step, which will inevitably be a step he takes at some point to the Premier League."