Kane has been subject of interest from Bayern Munich, Manchester United and Paris Saint-Germain this summer, with the Bundesliga club seeing two bids rebuffed.
Bayern claim Kane wants to join them but so far Spurs have deemed the offers for their all-time leading scorer as insufficient.
Kane is into the final 12 months of his deal and despite not pushing for a move away form the club, the 29-year-old is keeping all options open surrounding his future.
That means the England skipper could see out his final year at the club before leaving next summer as a free agent.
Spurs are desperate to tie Kane down to a new deal but, as yet, there have been no signs of a breakthrough.
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Postecoglou was quizzed about the ongoing uncertainty around Kane's future and revealed his unease at the situation.
"Fair to say I am not relaxed about it! It's not something you go: 'Ah, well, you know if it does or doesn't happen!' I mean it's a very important part of this football club, not just the team, but the football club," he told reporters.
"He is such a massive figure, and as I have said already, whilst my focus isn't on it on a daily basis because I've got other things I need to do, I know that every time I am talking to you guys, or whenever Harry is going to talk, that's the first question you are going to get.
"So, you've got to deal with it, and I think, for everyone concerned, we don't want to be doing it for too long. I don't think that is good for anyone. I don't think it is good for Harry, I don't think it is good for the club, because as laser focused as we want to be, you end up sort of repeating yourself along the way.
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"But the flip side of that is I don't want to put a deadline on it, because that adds even more pressure. You want these things to happen for the right reasons. The reality of it is he's still a contracted player at our football club, so that's the way I see him.
"It's not like his contract is ending on the 12th [August] and he's got to make a decision, he's got another year. So from that part, I am not relaxed but I am not putting pressure on him or anyone else at the football club, saying 'well we need to do this'. But I don't think deep down any of us want it to go on for too long."
The Australian also opened up on his relationship with chairman Daniel Levy who is tasked with convincing Kane to commit his future to the club. So far, Postecoglou admits having a good relationship with the Tottenham supremo.
"I guess whenever you go into a role like this you always know that relationships are going to help you as a manager to get the things you need to work the way you want," he said.
"There is always one or two, wherever I have been, key decision makers - and Daniel is no different to all the other ones I have worked with. They are very passionate about their football club. That's the one common dominator.
"They are all different sorts of personalities, but they are all very passionate about their football club and they want their football club to be successful. So you understand that and then, for me, it's about saying, what's that relationship going to look like? Again, I think what he wants to do is support me."
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