Talks between Spurs chairman Daniel Levy and Bayern executives commenced on Monday regarding the future of the England captain.
The Bundesliga champions are planning to continue talks as they look to land the striker, who's understood to be valued at £100m.
The German side had a £70m offer for the England captain rejected earlier this month and it's thought Bayern are willing to pay up to £86m.
Kane's wife has also reportedly been recently spotted looking round houses in Munich amid his heavy links with the German club.
Spurs and Bayern's representatives were due to meet last Friday, but the face-to-face meeting was postponed until Monday.
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However, Spurs fans now look set to have to carry on waiting to see whether the striker will be with them for the 2023/24 season, with less than a month until the new campaign starts.
Kane has featured heavily in Tottenham's pre-season, most recently scoring a penalty against Singaporean club Lion City Sailors.
When speaking about the possibility of losing him this summer, Spurs manager Ange Postecoglou said: "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 he's a very important part of this football club, not just the team, but the football club.
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"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."