Greaves began his career with Chelsea before spells at AC Milan, Spurs and West Ham.
He scored 44 goals in 57 games for England and was a member of the 1966 World Cup-winning squad.
He holds the record for most goals in the English top flight with 357, and scored 266 in 379 matches for Tottenham.
Greaves’ 41 goals in 1960-61 remains a record in a season for Chelsea, and he also holds the Spurs record with 37 in 1962-63.
A Tottenham statement read: “We are extremely saddened to learn of the passing of the great Jimmy Greaves, not just Tottenham Hotspur’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.”
England captain Harry Kane, Spurs’ second highest goalscorer of all time second behind Greaves, said he was a “true legend”.
Former Spurs striker Gary Lineker described Greaves as a “giant of the sport”.
He tweeted: “Quite possibly the greatest striker this country has ever produced. A truly magnificent footballer who was at home both in the box and on the box. A charismatic, knowledgeable, witty and warm man.”
Former England defender and Tottenham captain Gary Mabbutt told BBC Radio 5 Live: “People talk about legends sometimes very loosely. With Jimmy, you can use that word consistently because he was that wherever he played – his sublime goalscoring ability, his finishing and the character that he was.”
Tottenham said Greaves possessed “immaculate ball control, great balance and such composure in front of goal that he rarely spurned an opportunity”.
He scored a hat-trick on his Spurs debut against Blackpool in December 1961, in the FA Cup final win in 1962 and twice in the Cup Winners’ Cup win in 1963 as they became the first British club to win a European trophy.
“Football will not see his like again,” Spurs said.
Greaves is survived by his wife Irene, four children and 10 grandchildren and great-grandchildren.
Greaves scored 132 goals in 169 matches for Chelsea between 1957 and 1961 before moving to Milan.
Chelsea said they “mourn the loss of a truly remarkable player and one of our own”.
He scored nine goals in 14 Serie A games for Milan, including on his debut, but did not settle in Italy and returned to London with Tottenham.
He left Spurs in 1970 to join West Ham before retiring at the end of the 1970-71 season at the age of 31.
The Hammers will hold a minute’s applause before Sunday’s Premier League game against Manchester United.
After retiring, Greaves moved into broadcasting and presented ‘Saint and Greavsie’, a football show with former Liverpool striker Ian St John, between 1985 and 1992.
Greaves suffered a minor stroke in 2012 and a severe stroke in 2015. He was awarded an MBE in the Queen’s New Year Honours list in 2020.
‘Admired by all who love football’
Greaves is fourth on the list of all-time England goalscorers, behind Wayne Rooney (53), Sir Bobby Charlton (49) and Gary Lineker (48).
He originally missed out on a 1966 World Cup winners medal because he was injured during the group stage.
After recovering, Greaves was left out of the starting XI for the final and his replacement, Sir Geoff Hurst, scored a hat-trick as England beat West Germany 4-2 after extra time.
The World Cup squad players and the families of manager Sir Alf Ramsey and other backroom staff were finally presented with medals in 2009.
England manager Gareth Southgate said his team will pay tribute when they play Hungary at Wembley on 12 October.
“Jimmy Greaves was someone who was admired by all who love football, regardless of club allegiances,” said Southgate.
“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.”