McNeill, who spent the whole of his playing career with the Glasgow giants, also went on to manage Celtic.
McNeill had suffered from dementia for a number of years and a statement issued by his children on the club's website said: "It is with great sadness that we announce the death of our father Billy McNeill.
"He passed away late last night (Monday, April 22) surrounded by his family and loved ones. He suffered from dementia for a number of years and fought bravely to the end, showing the strength and fortitude he always has done throughout his life."
McNeill joined Celtic from junior side Blantyre Victoria in 1957 and the defender went on to make a club record 822 appearances for the Hoops over 18 seasons.
He also won 29 caps for Scotland.
But it was arguably as the captain of the Celtic side that overcame the notoriously defensive Inter Milan 2-1 in the 1967 European Cup final in Lisbon that he will be best remembered.
McNeill enjoyed two spells as Celtic manager and, in total, won 31 trophies during his lengthy association with the Parkhead club.
"While this is a very sad time for all the family and we know our privacy will be respected, our father always made time for the supporters so please tell his stories, sing his songs and help us celebrate his life," the statement added.
McNeill was the captain of the Celtic side under celebrated manager Jock Stein that dominated Scottish football in the late 1960s and the early 1970s, winning nine league titles in a row, seven Scottish Cups and six League Cups.
He retired as a player in 1975 but returned to the club for two stints as manager, from 1978-83, and 1987-91 with spells at Manchester City and Aston Villa in between.
Even after his second term as manager he retained a strong link with Celtic, becoming a club ambassador in 2009 and six years later a statue of McNeill holding aloft the European Cup was unveiled outside Parkhead.
'Great Celtic man'
Paying tribute to McNeill, Celtic chief executive Peter Lawwell told the club's website: "This is the saddest of days for the Celtic family, and also for the wider football world.
"We mourn Billy McNeill's passing and we send our thoughts and prayers to his family and friends, while we also give thanks for Billy's life and the incredible joy he brought to so many people as a Celtic player, a Celtic manager and a great Celtic man."
Neil Lennon, the current Celtic manager, added: "This is such sad news and I want to send thoughts and prayers to Liz (McNeill's widow) and all the family from myself and all the backroom staff here at the club.
"Billy was a brilliant Celtic player, and a great manager, and as a player and a fan myself, it was always a great experience whenever I would meet him and chat to him."
Meanwhile McNeil's former team-mate Jim Craig tweeted he was "heartbroken over the passing of my great friend and captain Billy McNeill".
McNeill's managerial career also included a stint at Manchester City and the English side tweeted: "We are sad to learn of the passing of Billy McNeill who managed the Club between 1983-86.
"Everyone at Manchester City would like to send our sincere condolences to Billy's family."