Australian Postecoglou is set to be appointed as Spurs boss imminently but Gunners legend Keown is unsure whether the Celtic boss will be able to turn things around.
Last weekend Postecoglou guided Celtic to a domestic treble and the 57-year-old has been tipped to take the reins at the north London club which would see him replace Antonio Conte, who was sacked in March.
On Postecoglou's impending arrival at Spurs, Keown told talkSPORT:
"I want to have respect for all managers, but I thought [Antonio] Conte was an outstanding manager.
"That was my biggest fear actually [that Spurs would win trophies], because he was capable of doing special things.
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"Daniel Levy in 2021 was talking about a brand of football that he was looking for, whether or not this appointment is about that, whether or not he wants to see attractive football.
"Of course he wants to see a trophy being won, but it doesn't really strike any fear into me no, not as Conte did.
"He was a proven winner, of course he arrived here seven years ago [at Chelsea] when Pep [Guardiola] did [at Manchester City] in that first season, when he was without a trophy.
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"He was very pragmatic, laid down the law very quickly and they wasn't able to follow."
Keown added: "I don't know if the Spurs fans have ever been as frustrated in their history, so it's a difficult one for him [Postecoglou] to come into.
"Has he got the pedigree? He's going to walk into something he has never experienced ever before."
Simon Jordan, though, believes it could prove to be a shrewd move, suggesting he there's potential for Postecoglou to have a similar impact to what Jurgen Klopp had at Liverpool.
"I think it's a decent appointment," he said.
"I think he's a good manager, he'll play a brand of football that will appeal to Tottenham fans.
"He's coming into a league which is very competitive, but this is a man who has gone into Scottish football last year and scored 114 goals in 38 games so three goals a game. He can only beat what's in front of you.
"He's going to get a better group of players, arguably. Maybe not in mindset but in technical ability at Tottenham. I have this real admiration for the Australian outlook.
"When you get a decent Australian working for you you get a character and an ability to manage up, manage sideways, manage down. They have a desire to overcome obstacles rather than use them as an excuse.
"If he goes to Tottenham I think he'll be alright. I don't know what alright looks like for Tottenham because for Daniel Levy it means top four, it means the economics and it must mean winning something. You can't make a case for Tottenham winning much very soon.
"I think Postecoglou has the chops to work with Daniel, which is not an easy feat because I don't suspect and I've never made the case he's easy to work with.
"But I do not think Postecoglou is in any shape or form beneath Tottenham and I wouldn't be surprised if he gets them competitive. Will he win something with them? He might win a cup like Conte should have done.
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"I think it's great for him. He's got the character to galvanise and bring the fan base with him. It's somewhat similar to the way Klopp did with Liverpool.
"He's an inclusive manager, he doesn't seek to alienate or make little islands people sit on. He will need to have the intrinsic value of everyone being together and get them going."