Kane hailed Greaves, the only man to have scored more goals for the north London club, as “a true legend”.
“RIP Jimmy Greaves. A true legend and one of the great goalscorers,” the England captain tweeted.
“Thoughts are with his family and friends.”
Greaves scored 266 goals for Tottenham in 381 appearances for the club.
The striker began his career at Chelsea before a brief spell in Italy with AC Milan.
He was signed by Bill Nicholson to join the double-winning Tottenham side in December of 1961 for a fee of £99,999
Welsh wing wizard Cliff Jones, a member of that league and cup winning side, has also hailed his long-time teammate.
The pair formed a potent attacking partnership and helped power Tottenham to the 1962 FA Cup, a triumph they followed with victory in the European Cup Winners’ Cup one year later.
— Cliff Jones (@Cliff_Jones11) September 19, 2021
“Happy memories with my friend & teammate Jimmy Greaves,” Jones posted on Twitter. “Condolences to Irene & family, love Cliff , Joan & all the Jones family xx”
Ian Wright meanwhile revealed that Greaves was the first player that the former Arsenal striker was encouraged to mould his game upon.
Wright tweeted: “The first footballers name I ever heard from my teacher. ‘No Ian! Finish like Jimmy Greaves!’ May he rest in peace.”
Greaves scored 44 times for England in just 57 appearances for his country.
He was cruelly denied a role in his country’s 1966 World Cup Final zenith, an untimely injury ruling him out of the quarter-final.
Geoff Hurst took Greaves’ place and held on to the shirt despite the Tottenham man’s return to fitness, starting the deciding game against West Germany at Wembley and producing a decisive, famous hat-trick.
England manager Gareth Southgate confirmed that his side will remember Greaves at their next home fixture against Hungary in October.
“Jimmy Greaves was someone who was admired by all who love football, regardless of club allegiances,” said Southgate.
“I was privileged to be able to meet Jimmy’s family last year at Tottenham Hotspur as the club marked his 80th birthday. My thoughts are with them and I know the entire game will mourn his passing.
“Jimmy certainly deserves inclusion in any list of England’s best players, given his status as one of our greatest goalscorers and his part in our 1966 World Cup success.
“We will pay tribute to his memory at our home match with Hungary at Wembley Stadium next month. His place in our history will never be forgotten.”