The former Blackburn and Newcastle United striker netted 260 times in a career spanning 15 seasons in the reformed English top flight.
Wayne Rooney is his closest rival with 195 while Kane recently scored a hat-trick, against Stoke City, to bring his tally to 66.
Kane is aiming to score over 20 Premier League goals for the third successive season and Shearer is confident the 23-year-old can eclipse his record in the years to come.
"Harry Kane is good, but he's not that good," Shearer, speaking via the Premier League's Twitter page, joked
"He has an excellent chance if he stays fit and stays in the Premier League then, yes, he will have a decent opportunity of breaking my record."
Up next for Kane is Sunday's visit of Everton as Tottenham look to further tighten their grip on a top-four place and put pressure on Premier League leaders Chelsea.