Kane scored a hat-trick in Spurs' 5-2 win over Southampton on Boxing Day, taking his tally to a Premier League-record 39 in 2017, breaking a mark previously held by Shearer.
The England international's performances have seen him linked to Spanish and European champions Real Madrid previously.
Shearer believes that could be a problem for Tottenham going forward, particularly if Kane is desperate to win trophies.
"Not only has he beaten my calendar-year record in the Premier League, but with his second successive hat-trick he eclipsed Lionel Messi and Cristiano Ronaldo for goals in all competitions in 2017 for club and country," Shearer wrote in The Sun .
"That is some company to be in but, believe me, he could score goals for Barcelona and Real Madrid as well. There lies the problem for Spurs going forward, of course
"The club will not want to move into their brand new stadium without their biggest star.
"'One of our own' as the Spurs faithful chant.
"But things have stalled a bit on the pitch after title tilts in the last two seasons, although they have been impressive in the Champions League."
Kane is on track to win a third consecutive Premier League Golden Boot, something only Shearer (1994-97) and Thierry Henry (2003-06) have done previously.
However, Shearer said the 24-year-old – who is contracted until 2022 – could move unless Tottenham were successful.
"If they don't win anything in the next two years, Kane could have a big decision to make and there will be no lack of suitors with big money wanting him," he wrote.
"The fact Spurs are his club will be a big factor.
"I know as well as anyone what it means to play for your home-town club and stay loyal to them."