Greaves, 75, is being treated in intensive care after being taken ill on Sunday.
"Everyone at the club wishes Jimmy Greaves a speedy recovery after he suffered a stroke yesterday," Tottenham said in a statement. "Our thoughts are with Jimmy and his family at this difficult time."
Having established himself as a prolific striker with Chelsea for whom he scored a then club record 124 league goals, Greaves moved to Tottenham in 1961 and scored 266 times in 379 appearances -- a record that still stands.
He won 57 England caps, scoring 44 times and is fourth on the all-time scorers list behind Bobby Charlton, Gary Lineker and Wayne Rooney.
Greaves was part of England's 1966 World Cup winning squad but was injured in the group stage and replaced by Geoff Hurst.
He originally missed out on a medal because they were only awarded to the 11 players who featured in the final, although he finally received one in 2009.
Geoff Hurst, who replaced Greaves, said on Twitter: "Thinking of my friend and one of our greats this morning, wishing him all the best and a speedy recovery."
Messages of support also came in from Chelsea and West Ham United, another of Greaves's former clubs.
Greaves, who also had a short spell with AC Milan, was at his most deadly while with Tottenham and is still revered there.
He scored in the 1962 FA Cup Final and 1963 European Cup Winners Cup Final and added a second FA Cup in 1967.
Greaves, who had a minor stroke in 2012, is due to be inducted into the club's Hall of Fame on May 13.
A message was posted on Greaves' Twitter account saying: "We have had thousands of get well messages including one from a devastated (former Scottish footballer) Ian St John. Thank you to everyone for your thoughts at this time."
After his playing days, Greaves and St John hosted a popular football show on ITV.
(Reporting by Martyn Herman, editing by John O'Brien and Pritha Sarkar)
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