Cox, 36, admitted that she has not been approached by ITV about working on either format, but joked that she wouldn't let being "tone deaf" get in the way of her taking part.
"If Simon's reading this, my decade of playing pop on the radio to the nation obviously gives me the right to judge others," the mum-of-three told The Mirror.
"The fact I'm completely tone deaf and struggle to know if something is out of key is irrelevant. I'd just be like, 'That's brilliant!'
"The X Factor and The Xtra Factor are great shows so it'd be amazing to do it. But unless they ask me at some point in the next two weeks, I haven't got it!"
She also backed Cheryl Cole to be a huge success in the US, describing the Geordie popstar as a "sweetheart".
Cox joins a long list of potential Xtra Factor hosts including Kimberley Walsh, Matt Edmondson, Georgie Thompson, Laura Hamilton and Jeff Brazier.
> Matt Edmondson hoping for 'X Factor' return