His interest in the Hoops' comes down to the fact his cousin - once removed - Eoghan O'Connell is a fast-rising star at the club.
The younger O'Connell made his first-team debut as a substitute for Celtic in their 3-1 Antalya Cup semi-final victory over Trabzonspor earlier this month - and then had his first start in the final against Galatasaray a few days later.
The central defender's father Damien is Paul's first cousin, but he has a rather different sporting background to the Irish rugby legend.
"He's from a GAA family so he would have grown up playing hurling, gaelic football and soccer," O'Connell said. Damien played for Cork City and Cobh Ramblers, who are the team Nottingham Forest signed Roy Keane from.
"Eoghan would have grown up watching his dad playing for them so it was hurling, gaelic football and soccer all his career. I don't know if it's in the genes, but it's a sporting family. The good thing about it is that Irish kids play so many different sports it probably all helps."
Eoghan, 18, was an unused substitute for Celtic in their 4-0 SPFL victory over Kilmarnock on Wednesday evening.
Paul, 34, who has played 88 Tests for Ireland and seven for the Lions, knows what it takes to reach the top, but said he was keen to see his young relative come through.
"I'm pleased to see him develop his career at Celtic," said the Munster and Ireland captain. "I'm delighted. It's going to be a tough journey for him at Celtic. We're hoping he makes it through because I'll be retiring in about two years and I'll be able to get over to Glasgow and Parkhead every second weekend to watch him play and bring a gang of the family over.
"I'm not a Celtic fan and I've never been to a game. I'm actually an Everton supporter but whenever we play Glasgow over there Eoghan always comes over to watch our games and I meet up with him. He's a great lad and it will be nice to go and watch him play.
"I was delighted, absolutely delighted, when he made his debut. I did not see the game but have a recording at home because it was on Eurosport. I am looking forward to getting weekends away in Glasgow watching him."
One video the rugby-playing O'Connell has watched a lot of recently is Ireland's agonising 24-22 loss to the All Blacks last November, when a last-minute try by Ryan Crotty, New Zealand's replacement full-back, denied what would have been one of the most famous victories in Irish sporting history.
"Joe [Schmidt, the Ireland coach] wanted to show us opportunities that we had to stop them from scoring the try," O'Connell explained. "When you are playing a game of that intensity, it is about being mentally tough and digging in because it was probably tiredness where a lot of the issues came from.
"That's what he was just saying, that when big games like that are on the line you need to dig in and make sure. He was just showing us, we were watching it, unfortunately very slowly, and he was able to highlight some very simple bits of play, things, as I said, that did not need to be so spectacular - we could have stopped them scoring that try."
However, O'Connell dismissed the idea that Scotland would feel the backlash from that game when they take on Ireland at the Aviva Stadium on Sunday. "There is frustration from the day, but I don't know about channelling it into the Six Nations," he said. "It's more about the level of performance we achieved that day and making sure that that's more or less our standard performance.
"That's the thing for us. At times, we've been inconsistent in Ireland. One of the things that Leinster have had is their consistency and Joe [who coached Leinster before taking the Ireland job] played a big part in that in the way he prepared teams and the way he urged players to prepare mentally.
"Hopefully, we can address that inconsistency in the Six Nations."