Former players Cristiano Ronaldo and David Beckham were quick to lend support.
Ronaldo said: "My thoughts and prayers are with you, my dear friend. Be strong, Boss!"
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"Keep fighting Boss.. Sending prayers and love to Cathy and the whole family x," wrote David Beckham on Instagram.
A post shared by (@davidbeckham) on
Manchester United captain Michael Carrick said he was "absolutely devastated", adding: "Be strong Boss".
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The club's Spanish goalkeeper, David De Gea, tweeted two prayer emojis for his former manager.
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Bryan Robson, who played under Sir Alex until 1994 and remains Manchester United's longest serving captain, told Sky Sports News about his shock at the news.
"Jason, his son, gave me a call last night. So it did come as a big surprise and a shock. From what I've had the operation went well and the boss will make a full recovery and that's what everybody would wish for.
"Everybody respects what he did in the game. Everybody thinks its all this hair dryer treatment. He enjoys his racing, he always has a good time and he's good company when you are around him."
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Bryan Robson: Sir Alex's operation went well
Those who played under Sir Alex, including Wayne Rooney and Peter Schmeichel, also posted to social media in support of their former boss, along with stars of the current team, such as Jesse Lingard and Romelu Lukaku.
Former Arsenal stars Thierry Henry and Robin Van Persie - who played under Sir Alex in his final season - wished him a speedy recovery, as did former footballer and pundit Gary Lineker.
Arsenal player Mesut Ozil posted to Twitter in support of Sir Alex, as did former Chelsea striker Didier Drogba.
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The Red Devils' rivals Manchester City said they wished Sir Alex a quick recovery, adding the hashtag #footballfamily.
In a statement on Liverpool's website, a spokesperson said Sir Alex was "a great rival but also a great friend".
"The thoughts of everyone at Liverpool Football Club are with Sir Alex Ferguson and his family following the news that the former Manchester United manager is ill in hospital," they said.
"A great rival but also a great friend who supported this club during its most difficult time, it is hoped that Sir Alex will make a full recovery.
"In the meantime, the club will offer its full support to Manchester United and also his family."
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Aberdeen, the club he managed before his managerial move to England, sent its "thoughts and prayers" as did Tottenham Hotspur.
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Other clubs including Paris Saint-Germain, West Ham, Chelsea and Everton have also sent their wishes.
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Golfer Ian Poulter shared a photograph on his Twitter page of him sitting side-by-side with Sir Alex.
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Television presenter Eamonn Holmes, a longtime friend of the football hero and Manchester United fan, described him as an "amazing man".
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