The 29-year-old has managed 213 goals in the English top flight, and is bearing down on Shearer's all-time record of 260.
Kane has one year left on his current Spurs deal, with Tottenham recently rejecting a £60million bid from Bayern Munich.
Manchester United and Real Madrid have also been linked with the England international, who is valued at £100m by Spurs.
The north London club remain adamant Kane is not for sale, and are hopeful he stays. Shearer thinks differently, however.
"If Harry wants to join Bayern, I'll drive his f****** car there myself; anything to protect my Premier League goalscoring record of 260 - Harry is on 213," Shearer wrote in a article for The Athletic.
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"In truth, any hope there is probably forlorn.
"Harry is nearly 30, but he looks after himself, he's steered clear of major injuries and he's perfectly capable of coming back to England in a couple of years' time and banging in enough goals to overtake me.
"Still, though. Give me the keys and a sat nav and off I'll go…"
Last campaign, Kane overtook Wayne Rooney's record of 208 Premier League goals, and needs 47 more to surpass Newcastle legend Shearer.
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It remains to be seen whether Kane will remain at Tottenham next season, or move elsewhere.
A move to Bayern could see the England ace pick up a title or two along the way, something which has so far evaded him in his career.