Their lawyer Peter Ross told US media earlier this week that they had been "in negotiations for a month to find a resolution" with the 'One Thing' group's team.
Speaking to BBC Newsbeat, Sean O'Leary said that Cowell is "obviously a very smart man", but warned: "We are not intimidated. The law isn't a popularity contest. Our name has been infringed.
"We are obviously not as successful as the UK band. We're nothing like them either, we're more rock and play instruments. We put everything into our band. Our music is our lives."
Explaining the origins of the five-piece's name, he recalled: "When I was small and tried to start things up, my dad would always advise me you're at the bottom so there's only one direction to go. And that was the attitude I needed for the band."