The 52-year-old, who won two league titles, the FA Cup and the European Cup Winners' Cup at Everton in the mid-1980s, revealed the news via a statement on the club's official website.
Sheedy said: "It’s not ideal but I have to remain positive. There is a history in the family."
The former Republic of Ireland midfielder, who made more than 350 appearances during his 10 years as a player at Goodison Park between 1982 and 1992, also called for those who have any concerns about their own health to seek help immediately, and expressed optimism that the early detection of his own condition will see him back on the training pitch before too long.
"It’s so important that people don’t ignore the early signs and symptoms and don’t be afraid to consult a GP," he added.
"After a period of rehabilitation I will return to my coaching role at Everton."