A statement from Greaves' family confirmed that the 75-year-old had been taken ill, having recovered from a minor stroke in 2012.
Greaves scored 44 goals in 57 England games, the fourth-highest total of all time.
He also scored 266 goals in 379 matches for Spurs.
A statement issued on behalf of Greaves' family read: "Football legend Jimmy Greaves suffered a severe stroke on Sunday May 3, and is in intensive care in hospital.
"His wife, Irene, and four children have asked for privacy during what is a worrying time for the family.
"They will issue a bulletin on his condition when it is appropriate.
"England's greatest ever goal scorer was 75 on February 20, and has been working regularly on the personal appearances circuit in harness with his friend and agent Terry Baker.
"Jimmy had a minor stroke in 2012, but appeared to have made a full recovery. He was due to be inducted into the Tottenham Hotspur Hall of Fame on May 13 at a sell-out ceremony at White Hart Lane.
"This is all we wish to say at the present time."