The 27-year-old would reportedly prefer to stay in the Premier League, with Chelsea, Manchester City, and Manchester United considered the most likely destinations. City and United appear to see Kane as an alternative to Borussia Dortmund's Erling Haaland.
It's unclear if Spurs chairman Daniel Levy will let Kane go, especially to a rival in the same league. Levy apparently values him at around £150 million.
However, Spurs insist the striker isn't for sale, according to The Athletic's Jack Pitt-Brooke.
The Athletic also reported in April that Kane would demand a transfer if Tottenham failed to make the Champions League. Though they're not mathematically eliminated, Spurs remain a distant five points behind fourth-placed Chelsea with just two matches remaining.
Kane has made it clear he wants to win trophies, and that desire is intensifying as he enters his prime. He's yet to win a title in seven years with the club, a drought that continued with last month's defeat to Manchester City in the League Cup final.
"Individual awards are great. They're fantastic achievements. At the end of my career, these are things I will go over and take in a little bit more. But the goal right now as a player is to win team trophies," Kane said last month. "As much as this is great, I want to be winning the biggest prizes there is to offer as a team, and we're not quite doing that."
Interim manager Ryan Mason later suggested Kane's future doesn't depend on Champions League qualification.
"Harry Kane loves this football club. That's been proven over the last seven or eight years. He's one of the best strikers in the world," Mason said. "Everyone knows that, we know that, and I'd like to think Harry knows that as well."