The 69-year-old announced on Liverpool's official website that he is in the early stages of the condition following hospital tests.
"I've got to get on with it and I will. It's the way I've been brought up. Nothing has come to me easily," McDermott told liverpoolfc.com.
"I'm not frightened of taking it on and also, as we've seen, there are a lot of former players in a worse state than me."
McDermott's announcement comes just two days after Manchester United great Denis Law, 81, revealed he has been diagnosed with "mixed dementia".
"Battling is second nature. The worst thing was, until my condition was diagnosed you don't know what's going on," added McDermott.
"The number of ex-players being diagnosed with dementia or Alzheimer's is frightening."
McDermott made 329 appearances and scored 81 goals for Liverpool between 1974 and 1982.
He helped the club lift four league titles and the European Cup three times as well as the UEFA Cup twice.
McDermott also played 25 times for England.
