The England, Tottenham and Chelsea great was taken to a hospital near his Essex home on Tuesday and is understood to have had a comfortable night and with further tests to come this morning.
A further update on the 80-year-old, who has been wheelchair-bound since suffering a severe stroke in May 2015, is expected later on Wednesday.
Greaves' former Tottenham teammate Cliff Jones, who has credited the prolific striker with saving him from alcoholism, was among those to wish him well.
Jones wrote on Twitter: "Wishing my dear friend and teammate @jimmy_greaves a speedy recovery. Love from the Jones family x".
Greaves' agent Terry Baker, who runs his official social media accounts, wrote on Facebook last night: "If you love him like I do, have good thoughts for him as you go to bed tonight. He is my biggest hero of all time...
"He has endured a very difficult five years since his massive stroke in 2015 so spare some time to think of my lovely mate tonight."
Nobody has scored more top-flight goals in English football than Greaves, who began his career at Chelsea before spells at AC Milan, Spurs and West Ham.
Part of England's 1966 World Cup-winning squad, Greaves also scored 44 goals in his 57 internationals between 1959 and 1967 and featured in two World Cups.
He played in England’s three group games in 1966 but was not recalled after a shin injury ruled him out of the quarter-final, with his replacement Geoff Hurst going on to score a hat-trick in the historic win over West Germany.
Greaves won two FA Cups with Spurs and the European Cup Winner’s Cup, and his goals helped Milan win Serie A in 1962, though he had moved to north London mid-season. With 266 in his 379 matches for Spurs, Greaves remains the club's greatest-ever goalscorer.
"Everybody at the Club sends their best wishes to Jimmy and his family," Spurs said in a statement on Tuesday night.