Bayern announced the news in a press release via their official website as they paid tribute to Muller ahead of his 70th birthday later this year.
Muller, who once held the record for World Cup finals goals, has been undergoing professional care since the beginning of February this year, the Bundesliga champions said.
Bayern chairman Karl-Heinz Rummenigge paid tribute to Muller.
He said: “Gerd Muller is one of the greats of world football. Without his goals, FC Bayern and German football would not be where they are today.
“Despite his success, he was always modest, and that’s what really impressed me. He was a wonderful player and a friend.
“He brought experience as a coach and helped to create world champions in Philipp Lahm, Bastian Schweinsteiger and Thomas Muller.”
Rummenigge also asked for Muller’s family to be given privacy over the coming weeks.
“This is not only the express wish of his family, but also FC Bayern,” he added. “Bayern will always support Gerd Muller and his family whenever it is necessary.”
Muller’s doctor, Professor Dr Hans Forstl, thanked the club for the support given so far.
“Despite unmistakable signs of his illness, he was treated with great sympathy and respect by the Bayern family, fans and the media,” he said.
“That was very important because it is a wish for every Alzheimer’s patient to stay in the environment in which they feel comfortable for as long as possible.”
Muller was part of the Bayern side that won three European Cups in a row between 1974 and 1976, as well as four Bundesliga titles. He scored a remarkable 365 goals in 427 Bundesliga games.
Muller also scored an incredible 68 goals in 62 appearances for West Germany, winning the European Championship in 1972 and the World Cup two years later.