Club ambassador Mabbutt said Spurs would do everything they could to hold on to their record goalscorer, whose contract has just over a year to run.
"We will do everything we possibly can that Harry will remain a Tottenham player for the rest of his career," Mabbutt told reporters in Bangkok.
"So that's what we're hoping for. That's what we believe is going to happen."
Kane has bagged 27 goals this season, second only to Manchester City's Erling Haaland.
But with Tottenham sixth in the Premier League and out of the hunt for Champions League places, he has been linked with a summer move to Manchester United or Paris Saint-Germain.
Tottenham are on their third manager of the season -- interim boss Ryan Mason -- and Kane has publicly criticised the club's culture since the sacking of Mauricio Pochettino, who led them to the 2019 Champions League final.
But Mabbutt, who lifted the FA Cup as Spurs skipper in 1991, insisted the turmoil would not put big-name coaches off taking the job.
"All the big names want the job," he told AFP.
"To be manager of Tottenham Hotspur football club is a massive role for any manager."
Spurs take on Leicester City in the Thai capital in July as part of a tour of Australia and Asia that also sees them play West Ham and former coach Jose Mourinho's Roma.
