Our greatest-ever striker remains in hospital after suffering a serious stroke on Sunday, May 3.
The update, posted on behalf of Jimmy by his agent Terry Baker on Greavsie's official Facebook page today, read as follows:
"Jimmy is improving still. It's a long hard road ahead but there is room for cautious optimism. Jim and the whole family want me to convey how much it means to all of us, that there have been hundreds of cards and gifts and flowers and many many thousands of messages of goodwill on social media. We would like to thank Tottenham Hotspur for a fabulous bouquet and their heartfelt messages and all the famous footballers of all eras and all the media people Jim has worked with down the years for all their love and offers of help. From the bottom of the hearts of all concerned and especially from Jimmy, a big big thankyou to you all."
Jimmy, 75, is our record goalscorer with 266 goals in 379 appearances between 1961-70.
He scored in the 1962 FA Cup Final and 1963 European Cup Winners Cup Final and added a second FA Cup in 1967.
A member of England's World Cup winning squad in 1966, Jimmy also scored 44 goals in 57 caps, placing him fourth in England's all-time scoring list.