The American band are suing Cowell's Syco record label and Sony for trademark infringement, and are seeking more than $1m (£630,000) in damages.
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."
O'Leary's bandmate Scott Nagereda said that they had been made aware of One Direction's existence when the British group were competing on The X Factor in 2010.
"It was just so awkward that someone with our name was on a game show," he said. "That name means much more to us. It's everything."
Addressing the "awful and upsetting" hate mail and death threats they have received since news of the lawsuit came to light, O'Leary claimed that "99%" had come from British fans.
"We're not trying to upset anyone. We just want to keep what's rightfully ours," he said.
A full version of the interview with the American One Direction will be broadcast on BBC Radio 1 at 12:45pm.
> One Direction lawsuit manager: 'We won't be pushed around'
> One Direction in trouble: Bands forced to change their name
Watch One Direction's 'One Thing' below: