The Australian is on a rolling one-year contract at Parkhead, and the Scottish champions are hoping to convince him to stay.
Postecoglou has made it two league titles and two League Cups in two years with Celtic, and is hoping to make this year a treble with a Scottish Cup final still to come.
Spurs have had a number of stumbles in the search to replace their previous full-time manager Antonio Conte.
Julian Nagelsmann and Arne Slot were both believed to be leading candidates before dropping out of the race, while much-loved former manager Mauricio Pochettino has joined Chelsea.
Links persisted to former Spain and Barcelona boss Luis Enrique, but it now seems Postecoglou could be ahead in the running.
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The 57-year-old Greek-born former defender was previously a candidate for Chelsea, and was recently asked about links to Spurs.
"I've been asked about these things plenty of times," he said. "What's important for me is the here and now. I've never planned anything in my football career.
"My ambitions right now are to enjoy tomorrow [Saturday], have a good game and then try and win a cup final and make it a special season.
"It's not every year you get the opportunity to win a treble and if anyone thinks I'm thinking anything other than that they don't know me."