Greaves had suffered a stroke in May 2015 which left him wheelchair-bound and with severely impaired speech. He had also suffered from alcoholism and quit drinking in 1978.
Greaves is the club's record goalscorer with 266 goals in 279 appearances and has been remembered as "finest marksmen this country has ever seen".
England captain Kane is second on the list, 43 behind Greaves, and has hailed the England World Cup winner's numbers.
"Frightening, really, how good a player he was, the goal ratios that he returned, the goals he scored year in, year out," Kane told Sky Sports.
"For someone like me to look at those numbers and try to achieve those numbers and hopefully one day go on to break those numbers would be incredible."
In announcing the news, Spurs said Greaves, who also played for Chelsea, West Ham and AC Milan, died at home on Sunday morning.
The Premier League club said in a statement: "We are extremely saddened to learn of the passing of the great Jimmy Greaves, not just Tottenham's record goalscorer but the finest marksman this country has ever seen. Jimmy passed away at home in the early hours of this morning, aged 81.
"Throughout his wonderful playing career, Jimmy's strike rate was phenomenal. His Spurs return was 266 goals in 379 appearances between 1961 and 1970 – 220 goals in 321 league games, 32 goals in 36 FA Cup ties, five in just eight League Cup ties and nine in 14 European matches."
Both Spurs and Chelsea remembered Greaves in their Premier League game at the Tottenham Hotspur Stadium on Sunday afternoon, which the visitors won 3-0, with the players wearing black armbands, while former team-mates including Martin Chivers and Steve Perryman were in attendance.
A minute's applause was held before kick-off in north London and prior to Sunday's other two top-flight games – West Ham v Manchester United and Brighton v Leicester.
Greaves scored 44 goals in 57 senior appearances for England – the fourth highest in Three Lions history. He was a member of the 1966 World Cup-winning squad, but famously did not feature in the final victory over West Germany.
A shin injury had seen him replaced in the side by Geoff Hurst at the quarter-final stage and, although he was fit for the final, Alf Ramsey's decision not to change a winning side was vindicated by Hurst's hat-trick on the day.