For the Leeds pair head into their second-round games at the Betway UK Championship, looking to become the first father-and-son combination to play each other at a ranking tournament in over 30 years.
The only other time it has happened was in 1986 when Neal Foulds beat dad Geoff 5-0 in the last 32 of the International Open.
Victories for Peter against China’s Xu Si, and Oliver against Northern Ireland’s Mark Allen, would secure a third-round showdown next week.
Peter – a 47-year-old Tour veteran after turning professional back in 1991 – believes facing his son, 22, at the York Barbican would not only be big news in the Lines household, but for snooker fans around the globe.
“I joked to Oliver, (World Snooker chairman) Barry Hearn should have a word with our two opponents and let us win, because the exposure would be fantastic if we played each other,” said dad Peter.
“I can guarantee we will both be trying 100 per cent because we both really want it to happen.
“Especially at the UK, which is so close to home, and people would love it. I don’t know whose corner (my wife) Sarah would be sitting in.
“This game has to happen eventually – even if it’s not in this competition – it will happen, if I stay on the Tour.
“It would be great if it could happen in such a fantastic place like York.
“I can’t get my head around what it would actually be like, to be on the opposite side of the table as Oliver.
“I don’t really want to think too far ahead, we have got games to win before that, and I don’t want to jinx Saturday.
“Hopefully we can do an interview on Saturday night and can really look forward to it. It would be fantastic.
“It would be a great story for the UK Championship.”
The pair are based at the Northern Snooker Centre in Leeds, but rising star Oliver trains frequently with Judd Trump at the Grove Academy in Romford.
And with both being professional sportsmen, Peter admits he is the ultimate “competitive dad”.
He admitted: “We don’t practice as much, as we used to do, because it’s really awkward.
“When I am playing well, no matter who it is, even if was playing (wife) Sarah, I would want to hammer them.
“If he is not playing too well, I like to give him advice. It’s difficult.
“But when he plays me, he just wants to thrash me.
“We played each other in a pre-am and he absolutely hammered me. There will be no quarter given. No-one will be able to accuse us of letting the other player win.
“Sarah doesn’t play, but we got my little son his first table for his second birthday.
“Leo is on the baize already, I am dropping down the rankings at home, never mind with World Snooker.”
Lines’s best results as a professional have come at the UK Championship, where he reached the quarter-finals back in 2009. Last year Lines – who won the 2016 World Seniors title – competed as an amateur, after falling off the professional tour.
But that did not stop him causing another shock in York, when he knocked out former world champion Neil Robertson.
And after winning back his professional Tour status, Lines is eager to add to his scalps, after beating China’s Zhou Yuelong – the world No 29 – 6-5 in his opening match.
“I wish I could play York every game, every competition,” said Lines, who reached the last 16 of this season’s European Masters. “I always seem to do well.
“I don’t know what it is, but the UK Championship just brings out the best in me.
“I have always done well in the tournament. My best ever run was in the UK, reaching the quarter-finals in Telford. I just really like the comp.
“Certain competitions you seem to do well in, and there’s no reason for it, because you are playing the same people all the time.”
Against Zhou Yuelong, the match was locked at 5-5, before Lines clinched a 6-5 victory.
“I was really pleased with the win because I have lost to him already this season,” said Lines. “He will win competitions in the future, he is a fantastic player.
“It was a strange game. I was in front all the time, never behind, but I felt like I was clinging on.
“I just kept saying to myself ‘stay in front’, because every frame he won, he won in one visit. I couldn’t shake him off, it went down to the wire. Luckily I got a chance in the final frame, and I won.”
Oliver has arguably the tougher test today, facing world No 9 Allen.
But Peter is backing his son to add to the Lines family upsets, when he cues off this afternoon.
“Mark Allen is a world class performer, so it will be a really tough game for Oliver,” said Peter.
“But Oliver is playing well. If he can go out and believe in himself, there is no reason why he can’t win.
“Mark Allen will have to play well to win.”
Peter will take time out from his own preparations, he plays this evening, to watch Oliver in action.
“I will be watching Oliver in the afternoon,” he said. “It’s a bit awkward for me, because I want to prepare for my own game. But I won’t think about that when I watch him play.”