But Kane has continued to deliver the goods, with the England international scoring 17 goals in the Premier League this term.
Worryingly for Tottenham, however, both Manchester United and Manchester City are on the lookout for a new No 9.
United have seen Anthony Martial misfire this term, while City are hunting for a long-term Sergio Aguero replacement.
And, now, Kane has plunged his Tottenham future into uncertainty by refusing to commit his future.
Asked if he’d have to move to win trophies Kane said: “I think that’s a hard question to answer right now.
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“To be thinking about speculation or rumours would be damaging in terms of my own performance.
“I always like to be focused on one goal and one job and that’s to finish strong with Spurs, win these qualifiers with England and hopefully go on and have a great Euros.
“Look, I try to stay out of that speculation as much as possible. I’m fully focused on doing the job on the pitch from now until the end of the summer and then we’ll see where we go from there.”
Kane also admitted it had been a ‘difficult’ campaign for Tottenham.
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“Obviously the Arsenal game was difficult to take – in a north London derby you never want to lose. And going out of the Europa League was a big disappointment.
“But we bounced back last Sunday with a win and I think when you look at the bigger picture we’re a couple of points off top four and we’ve got a cup final to play.
“So if by the end of May we’ve finished in the top four and won the Carabao Cup I think everyone will say it’s been a pretty good season.
“In my career and as a person I never get too high or too low. It’s part of football.
“That’s what I’m trying to do and I’m sure the team are trying to do as well.”
Kane insists he’s got more to learn, however, despite being one of the world’s deadliest strikers.
“I’m not the finished article by any means,” he said.
“I’m at a very good place at the moment and hope to get better and better over the next few years.
“You’re seeing that year on year, with the goals he scores and the contributions he has – and a lot of them have been in the last five years.
“I feel, especially nowadays, there’s no reason why you can’t be hitting your prime in your early 30s.
“Looking ast Lewandowski, Ronaldo, Messi – there’s lots of players coming into even better form when hitting their 30s. That’s what I want to do.
“I’m 27 and still feel I’ve got plenty more to learn and to improve. So, that only gives me more motivation to do it.”