The 75-year-old former Tottenham, Chelsea and West Ham star was admitted to hospital on Sunday following a severe stroke which has left him in intensive care.
His wife, Irene, issued a statement on Tuesday saying his recovery would be a long and slow process.
She said: "Jimmy is having to do what he is told, and understands the seriousness of what has happened to him
It is going to be a long, slow road to recovery and he is in the best possible hands.
"On behalf of Jimmy and our family, We would like to thank the many thousands of you who have voiced your support and concern
We would greatly appreciate it if the media would allow us privacy for what is going to be a long period of rehabilitation."
The world of football rallied around Greaves after it emerged he had been admitted to hospital on Sunday.
Gary Lineker, two places above Greaves in England's all-time top scorer list, said: "Choked to read that Jimmy Greaves has had a severe stroke.
"One of my absolute heroes
Thoughts are with him and his family
Get well, Jimmy."
Tottenham, West Ham, Chelsea and the Football Association all tweeted messages of support for the 75-year-old - as did his former England team-mate Sir Geoff Hurst.
"Thinking of my friend and one of our greats this morning, @jimmy_greaves
Wishing him all the best and hoping for a speedy recovery,'' Hurst said.
Greaves suffered 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.
He is regarded as one of the greatest English goalscorers of all time - he scored 44 goals in 57 games for his country and also netted a club-record 266 times in 379 matches for Tottenham.
After retiring from the game, Greaves co-hosted the popular Saturday lunchtime football show Saint and Greavsie alongside former Liverpool forward Ian St John.
Source : PA
Source: PA