His family say the 75-year-old is able to sit up in bed, adding that while he is "by no means out of the woods there is room for cautious optimism."
Greaves has a club-record 220 league goals for Tottenham from 1961 to 1970.
His 44 goals in 57 caps puts him fourth in the list of all-time England goalscorers behind Sir Bobby Charlton, Gary Lineker and Wayne Rooney.
A statement on Tuesday from Greaves's wife Irene read: "Jimmy is having to do what he is told and understands the seriousness of what has happened. He's in the best hands."
Greaves, who had a minor stroke in 2012, scored a then-club record 124 league goals for Chelsea between 1957 and 1961.
He also played for AC Milan and West Ham and was part of England's 1966 World Cup-winning squad but originally missed out on a medal because he was injured during the group stage.
In the 1980s he forged a new career as a football pundit on ITV, most notably alongside former Liverpool striker Ian St John on Saint and Greavsie, which ran from 1985 to 1992.
Ex-England and Tottenham striker Jimmy Greaves is out of intensive care after suffering a severe stroke on Sunday.