Tottenham chiefs are adamant their star striker, who has netted 15 goals in 13 games for club and country this season, is going nowhere.
But despite Spurs’ hardline stance, Allardyce believes a move could be on the cards in the next few years.
He told Paddy Power: “Harry Kane is clearly happy at Tottenham, at the moment. I think he’ll be looking at staying for the next two to three seasons but, after that, who knows? Maybe it’ll be time to move on.
“He could easily attract interest from the big guns – Real Madrid, Barcelona, and Bayern Munich.
“Those three clubs are the only three that would tempt Harry to leave Tottenham, especially if he wins something there.
“I don’t think Spurs would ever want to sell Harry Kane to a Premier League club. But they sold Gareth Bale in the end – as much as they want to hold on to him, money does talk.”
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Sam Allardyce thinks Harry Kane could be sold by Tottenham
“When you see what happened with Neymar going to PSG, there’s always an offer that someone can’t refuse, and I think that’ll be the case with Harry eventually.
“There are stories that Real Madrid want to offer £200million for Kane. Can Tottenham reject that kind of money? I don’t know.”
Kane signed a new five-and-a-half year deal in December 2016 that will keep him at Tottenham until 2022.
He became the first Spurs player to be paid more than £100,000-a-week and he remains the highest earner at the club.
Although he’s happy now, Allardyce is mindful that could change if he doesn't start picking up trophies.
Tony Gale: Harry Kane worth more than Neymar
He added: “It all depends in the end on Harry, if he decides that he wants to go. Kane doesn’t seem the type of player who is disruptive, a bit like Diego Costa when he wanted out of Chelsea, but he could be tempted by the likes of Real Madrid.
“He’s said that he wants to stay at Spurs until the end of the career, so presumably he thinks they can achieve what he wants to in the game.
“He’s obviously thinking they’re very close to achieving honours – they’re in the Champions League, they’re in their new stadium, he’s seeing the club grow. He’s enjoying it there at the moment.
“I think Kane plays as well as he does for Tottenham because he loves the club so much. There’s definitely a factor in that he’s so comfortable there, where he dreamed of playing as a kid.
“There’s a lot to be said for the idea that his performances are as good as they are because he loves that football club and gets on extremely well with the manager. He’s very comfortable with the manager, and feels he’s getting the best out of him.”
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