"As my accidental social media post indicates, I am undergoing treatment for cancer. However, unlike my mobile phone use, the treatment is going well," Sir Kenny said in a statement on Liverpool's website.
"This was not meant to be public, so I would appreciate it if the privacy of my family and me could be respected."
The Scottish international striker played 559 games for Liverpool between 1977 and 1990 and was also a coach at the Premier League club on two occasions.
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