The pair had a blazing stand-up row on The Late Late Show nearly eight years ago during a segment on selecting a song to represent Ireland in the Eurovision Song Contest, with the guitarist mentoring singer Laura O'Neill while Linda was on a panel of judges with Louis Walsh.
"It's gas, when it was first announced on social media that I was doing Dancing With The Stars, loads of people commented, 'I hope Linda Martin is not a judge' or 'I hope you're not partnered with Linda Martin'," said Billy.
"But look, that whole Linda Martin vibe, where she called me an odious little man and she said, 'Billy McGuinness you're not used to dealing with women with a brain'. Oh my God, how could she say that?
"When you think about it, The Late Late Show viewing figures went through the roof. Ryan Tubridy was chuffed. He said, 'Billy this is amazing, we are trending all over the place, it's the biggest viewing figures we've ever had'."
Billy has not met Linda since and laughs when asked if he'd do a waltz to her Eurovision-winning song 'Why Me?'.
"We would move in different circles and I never met Linda Martin up until that," he explains.
"I wasn't attacking Linda Martin, I was pointing out the fact that Louis Walsh shouldn't be on the panel because he managed Casey from Wonderland and he was also involved with Eoghan Quigg from the X Factor.
"So, to have him commenting on two singers that he was involved with, especially for us who had spent months trying to find a singer and a song to represent Ireland, I felt it was very unfair.
'And I stand by that. I still think I was in the right, that they shouldn't have had Louis because he was connected to two of the singers. It doesn't make sense.
"They could have had Niamh Kavanagh there, they could have had anyone that was involved in Eurovision, why did they have Louis Walsh there?"
Linda Martin admits she is a fan of Dancing With The Stars and has been asked to take part several times. However, she has always had to turn it down due to taking part in the likes of pantomime at the same time.
Speaking to the Sunday World last night, she again defended Louis's participation in the panel.
Of Billy, she said: "I haven't really thought about him much since. A lot of Eurovision fans abroad saw the Late Late clip afterwards and I had to explain to them what 'gobshite' was but I don't think anything about him now at all."
"I don't think it would be something Christy (Dignam) would be comfortable doing... I don't think Eurovision would be down his street." "If you said to me last year 'Billy you're going to be on Dancing With The Stars, I'd have went 'ah will you feck off, you're having a laugh', but here I am doing it. So, I'm never say never."
At the age of 61, Billy is the oldest contestant in this year's show and he points out he's paired with the youngest pro dancer, Hannah Kelly, who is aged 19.
Dancing With The Stars, tonight, RTÉ 1, 6.30pm