The American group are suing Cowell's record label over the use of the name, stating that they were around first.
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"We won't be pushed around," manager Dan O'Leary told The Sun.
O'Leary is demanding $1 million (£630,000) in damages, along with three times the profits made by the British One Direction so far.
"I don't care how powerful Simon Cowell is. He's mad if he thinks we're going to lie down, sit down or back down over this - whatever power and money he has behind him. We're not going to be pushed around by some music mogul," he said.
"The British One Direction have Mr Cowell's enormous resources behind them. We on the other hand do not. In our view, we were here first. We have rights, we have talent, and we have heart."
O'Leary, who is also the father of his group's lead singer Sean O'Leary, added: "My boy's dreams and the dreams of the band are just as important as the dreams of Cowell's group."
He explained that he had asked Simon Cowell to change the name of the British boyband before they began the US tour.
"It's black and white what's been done. We asked Simon Cowell to change their name before they came here. But they ignored us, and now we're reluctantly at this point of legal action."
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O'Leary continued: "The facts are that we were first to use the name 'One Direction'. We've been using it since November 2009 when, with big dreams and no money, we started to distribute demo CDs.
"The British band did not come together as a group until almost a year later. We were the first to record an album."
O'Leary insisted that he had no hard feelings towards the One Direction members themselves, and hopes that both bands can have success in the US.
"To Zayn, Harry, Niall, Liam, Louis, we wish you all well, and many years of success.
"There should be plenty of room for both bands to grow and thrive. We hope that this dispute will be resolved soon, and then all of us can focus on what we do best - making music."
The British One Direction recently appeared on an episode of hit Nickelodeon show iCarly in the US.
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Watch the video for '2012' by the US One Direction below: