The 'Promise This' singer was rumoured to be rejoining the show after being axed from the American version earlier this week.
Cole was given until midnight on Saturday to commit to the show ahead of Wednesday's auditions, but ignored Simon Cowell's calls as the deadline approached, reports the News of the World.
A source said: "The brutal truth is that no-one is bigger than the show - not Cheryl and not even Simon. We have bent over backwards for Cheryl and she hasn't engaged with us. She is out. That's showbusiness."
However, a close friend of Cole's said: "They set her a ridiculous deadline. They have basically done the same to Cheryl as they did to Dannii Minogue - given her an offer she couldn't accept.
"They did this because they knew Cheryl wasn't going to come back and they wanted it to appear that the decision was theirs. It wasn't - it was Cheryl's decision to go. There was a very slim chance she might have considered going back at a later date. But her heart wasn't in it. And once that deadline was set she was gone."
Manager will.i.am is also said to be "furious" with Cowell over the decision to replace Cole in the US with the show's co-host Nicole Scherzinger. Cowell, Paula Abdul and L.A. Reid remain as judges.
The British panel is now expected to be original judge Louis Walsh, Take That singer Gary Barlow, N-Dubz star Tulisa Contostavlos and former Destiny's Child member Kelly Rowland.
More on Cheryl Cole and The X Factor USA:
> Cheryl Cole and X Factor USA: Where it all went wrong
> Cheryl Cole pictures: From Girls Aloud to X Factor
> Knight: 'Cowell is a grand puppet-master'
> Paula Abdul: 'Cheryl Cole X Factor USA axe is news to me'
> Cheryl Cole to defect to UK 'The Voice'?
> Read our latest X Factor USA news