Kane spent two spells out with injury at Tottenham this season but still managed to score 24 goals in all competitions.
The England captain has been linked with a move to Real Madrid in the past.
But the La Liga giants have just signed another striker in Luka Jovic from Eintracht Frankfurt for £58million.
Burley believes Kane’s future will depend on what manager Mauricio Pochettino decides to do.
But he reckons the striker will stay in north London for another season at least.
“I can see Harry Kane going but not this summer,” Burley said.
“I think depending what happens with [Pochettino] will have a big bearing.
“Harry Kane will leave Tottenham, I’ll be surprised if that doesn’t happen, for big money.
“But I think another full season or thereabouts and full season fit and then Tottenham at some point will have to cash in.
“He’s going to force their hand at some point.”
On Pochettino, Burley says the Argentine will start next term at Tottenham but leave if he does not get things his own way.
“I think if he goes the cards start to come down a little bit and they collapse,” Burley said.
“I can see Pochettino doing another year.
“I can see some protracted conversations with Daniel Levy, maybe some promises about money, maybe some false promises.
“And then the season starts, he doesn’t get what he wants, gets annoyed and then looks to get himself out at some point.
“I think he will still be in charge when the season starts.”
Kane is set to be part of the England squad for this week’s Nations League matches.