Some close to Tottenham Hotspur believe it could come in the next few days. The striker will at least return to the club in that time, in what will naturally be perceived as a U-turn.
Not many think any of this will have any tangible effect on the situation.
The truth is that only one thing changes that. That is Manchester City putting in the kind of bid that Daniel Levy would find acceptable.
It is why, if the big question right now is how likely it is that Kane gets his move, many say the answer is that it just isn’t happening.
On City’s side, the hierarchy are currently considering what type of bid to submit. The club do have to balance Financial Fair Play requirements, and several sources say they can’t currently afford both Jack Grealish and Kane if it comes down to straight cash offers.
That might be different if City manage to offload a player or two, and they had previously made all of Raheem Sterling, Riyad Mahrez, Bernardo Silva, Gabriel Jesus and Aymeric Laporte available.
In a market where there is so little cash, though, all of the big clubs are struggling to sell players. It is why the expectation is that City will eventually submit a bid worth up to £150m, but with the value made up by players.
Two further factors complicate this, showing how far Kane is away from his dream move. One is that most of those players mentioned don’t want to drop out of the Champions League to play for Spurs. Why would any player willingly go? There are also questions over whether Spurs would even find them palatable.
When reports came out last week that the London club would agree to a sale, some senior figures appended laughing emojis to messages linking the stories. It was seen as preposterous.
Levy didn’t respond with laughter. He is said to have been furious, with his resolve to keep Kane only strengthened. The player is likely to be fined.
Many in the game have wondered about the tactics of the Kane camp throughout this.
One of the biggest questions is why his brother Charlie, who is his agent, hasn’t sought some outside help. It isn’t even necessarily a question of experience.
A super-agent with a lot of clients and influence, for example, could have gone to Levy with a lot more leverage. Agreements could have been struck about future deals. That’s how these agreements are generally struck. Personal favours and relationships are leaned on.
As it is, the Kanes have no leverage, beyond the notion of “a gentleman’s agreement”. That is just never going to trump the realities of contract law, and it remains baffling as to why the player agreed to a six-year deal in 2018.
It seemed woefully ill-advised at the time, no matter how good Spurs were under Mauricio Pochettino.
When it was announced, one elite coach, who has long coveted and admired Kane, quipped to a colleague: “Congratulations, you’ve just signed up to six years of winning nothing.”
It seems like this might have belatedly dawned on Kane, his ambition for trophies only fired by how close England got in Euro 2020.
It has left a situation where the striker didn’t show up to the first two days of training. These were tactics that were just never going to work with Levy.
It is not just the attempt to strong-arm him. There’s also the potential precedent. It might even make this the important refusal of Levy's entire time in charge.
Given how strong the Spurs chairman has been in terms of holding want-away players to contracts, as this situation brings to an extreme, it would set completely the wrong example to show that all they need to do is not turn up for training.
This is why Monday’s move was so baffling, particularly for a player seen as diligent and responsible as Kane.
To put it into context, it is not something Paul Pogba has ever considered – and that despite all of the comparatively unfair criticism he has received over his own desire to leave Manchester United. It should maybe change perceptions of both. Kane has certainly lost a lot of goodwill.
Monday’s move might have even upped any potential sale price by about £50m.
There’s another side to that, though. City obviously greatly rate Kane and would love him at the right price. It’s just that "the right price" might be some way off Levy’s exorbitant demands when you take into account the fact that the striker is 28, in this market. City could just be better off waiting.
There’s also the fact that Pep Guardiola now wants a straight number nine, but Kane is really a “nine and a half”, who actively seeks to drop back into the areas that players like Kevin De Bruyne occupy.
Would this really be worth such an outlay?
Are the potential gains really worth the events of the last few days, and maybe putting in a transfer request?
It doesn’t feel like it. Many supporters are already enraged. Levy is emboldened.
The problem is that, right now, nobody sees Kane being a City player at the end of this window.
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