Mike Sutton, who was 76, was also a former professional footballer and had suffered from dementia for the past decade.
He played for Norwich City, Chester City and Carlisle United from 1962-1972 before retiring due to injury.
He returned to football with Great Yarmouth Town in 1976, while working as a youth coach at Norwich.
In a touching tribute on Twitter, former Norwich, Blackburn, Chelsea and Celtic striker Chris wrote: "My dad Mike sadly passed away yesterday.
"He was a great sportsman, teacher, husband, dad and grandad. He was my hero and I will miss him so much."
Sutton, who starred in the Martin O’Neill era at Celtic Park, is now a successful television pundit and newspaper columnist.
Colleagues from BT Sport and the BBC posted their condolences on social media.
Football Focus host Dan Walker said: "Really sorry to hear about your dad Chris. I know you have been valiantly fighting his corner for a long time. May he rest in peace."
Rio Ferdinand posted: "Sorry to hear this Chris. Continue raising the awareness. Blessings to your family."
In November, Sutton, 47 – an outspoken campaigner for greater protection from head injury in football – spoke about his father’s fight with dementia.
He told ITV Anglia: "How he spends his days, the weight that he's lost, the fact that he can't go to the toilet properly - I defy anybody if that's a member of their family not to be heartbroken by that," Sutton told ITV News Anglia.
"But this isn't isolated. It's hundreds upon hundreds of players who've been neglected."
Earlier this month, Sutton claimed that football was “in the dark ages” when it comes to facing up to the consequences of head injuries.
He spoke out after it was confirmed that Sir Bobby Charlton is suffering from dementia.
Following his fellow World Cup winner Nobby Stiles’ death from Alzheimer’s. in October,, his family declared that football needs to "address the scandal" of dementia in the game.