McNeill, who spent the whole of his playing career with the Glasgow giants, also went on to manage Celtic.
Paying tribute to McNeill, current Celtic manager Neil Lennon told the club's website: "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."
McNeill had suffered from dementia for a number of years and a statement issued by his children 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 final of the European Cup -- "the trophy with the big ears" as he called it -- in Lisbon that he will be best remembered.
McNeill was at the heart of a brilliantly successful Celtic side assembled mainly from players in and around the Glasgow area by celebrated manager Jock Stein that dominated Scottish football in the late 1960s and early 1970s.
That team won nine league titles in a row, seven Scottish Cups and six League Cups.
McNeill retired as a player in 1975 and then had spells in management with two other Scottish sides in Clyde and Aberdeen.
He returned to Celtic for two stints as manager, from 1978-83, and 1987-91 with spells at English sides Manchester City and Aston Villa in between.
McNeill later became a Celtic ambassador and in 2015 a statue of him holding the European Cup aloft was unveiled outside Parkhead.
His fellow Lisbon Lion Bertie Auld said Tuesday that even as a teenager McNeill had made a major impact on his team-mates.
"I was there when the boss (Stein) signed Billy at 17 years of age and even then he had tremendous presence about him," Auld told Britain's Press Association.
"I remember in Lisbon, him carrying the ball out onto the park in Portugal. His chest was out and his head was back.
"He knew we would win and that filtered through the rest of us. He was never arrogant or egotistical. He just believed in himself and the rest of us."
Alex Ferguson may be best known for his hugely successful time as manager of Manchester United, but he had first-hand experience of McNeill's quality as a player when taking part in Old Firm derbies as a Rangers striker.
"He was a giant in Scottish football, a man with an incredible presence," Ferguson, who succeeded McNeill as manager of Aberdeen, told Celtic's website.
"He was also a truly good man and will be a loss to everyone who knew him. Farewell Cesar (sic)," he added, using McNeill's nickname.