In a wide-ranging interview for Faithful Through and Through, a book on the club and its culture to be published later this week, O'Neill also discloses that he believed Celtic and Rangers were on the verge of joining the English Premier League during his tenure as manager of the Parkhead club.
O'Neill, born in Kilrea, County Londonderry, 61 years ago, also insisted that Celtic should never relinquish its bond with Ireland.
He said: "The soul of Celtic Football Club is inextricably linked to its origins. Brother Walfrid's philanthropic influence pervades Celtic Park, his deeds and concern for the poor in Glasgow and the Irish itinerant workers moving into those areas are deeply rooted in Celtic's culture. That will never be erased. "
The former Nottingham Forest player took over as Celtic manager in 2000 and in five years won three Premier Leagues, three Scottish Cups and a League Cup. He praised his backroom staff of Steve Walford and John Robertson.
"The players had trust in Steve as a coach and John as an assistant manager. They knew that unless there were serious problems they weren't going to be reporting anything back to me," he said.
O'Neill's side won the treble in his first season, making him the only Celtic manager to achieve the feat apart from Jock Stein. The Northern Irishman said: "I was ecstatic to win it. It wasn't easy but nothing had been easy for Jock Stein."
He also pinpointed August, 8, 2001, as the moment his side grew up in Europe. "My view was that the winning of the game against Ajax was a very important breakthrough in our development as a side," he said of the 3-1 victory in a Champions League qualifier.
"It is worth bearing in mind the kind of players they fielded at that time, such as [Zlatan] Ibrahimovic, [Ahmed] Mido and Rafael van der Vaart. That night was a magnificent victory. The whole team was outstanding."
O'Neill guided the team to the UEFA Cup final in Seville in 2003 and addressed accusations that the club should have used that moment to take a further financial leap in acquiring players. "I think a lot of things changed; there wasn't any money - Celtic and Rangers could have gone their own way in terms of TV money and that might have earned them more. I know that suddenly there did not seem to be any money in the SPL in 2003 and 2005." He said that "in essence" Celtic and Rangers had stopped signing players for big fees.
He opened up the regular debate about an entry to English football for the Glasgow sides. "Celtic and Rangers may have to ply their trade elsewhere," he said. "It nearly happened when I was manager and I was all for it. Celtic and Rangers would maybe have to start in a lower league but what is five or six years in the life of a football club?"
Of his period in Glasgow and the atmosphere at old Firm matches, he said: "Personally I feel it would be strange for me to go to Ibrox and not hear Derry's Walls or not feel a presence at Celtic Park with Fields of Athenry resounding from all four corners of the stadium. Anyway, I often ask what is right and wrong these days.
"I totally accept that no-one wishes to take family and friends to football games and feel lives are under threat. Will viewpoints change massively or even at all in another 100 years? I really don't know the answer but I suspect not."
O'Neill famously led Lennon, the Celtic midfielder and now manager, towards the support after a controversial match in November, 2004. "I felt we were getting a rough deal that day," he said. O'Neill believed Peter Lovenkrands, the Rangers forward, had tricked the match official, Kenny Clark, into giving Alan Thompson a red card. "It seemed the referee did not need much persuading to send him off."
Chris Sutton, the Celtic striker, was also dismissed and O'Neill strode towards the Celtic support at the end of the match with his arm around Lennon. "That particular gesture at Ibrox was about appreciating the support we had from the Celtic fans that day," said O'Neill. "I think the referee was looking for any excuse to do it [the sending-offs]"
O'Neill was vocal in his praise of Lennon, saying: "He was in every way a leader. He proved himself time and time again on the European stage . . . He could be become one of Celtic's all-time greats."
Faithful Through and Through, by Richard Purden, is published by Headline at £20