Osbourne threatened to leave after two of her contestants were voted worst on Saturday's show, with Kimberley Southwick being ejected.
She told Dermot O'Leary she had failed: "What I did was no good, and I think I'm going home now, Dermot."
The News of the World reported that if she sticks to the claim she could be replaced by Moyles.
A source told the newspaper: "At this point in time Sharon has said she is no longer a judge on The X Factor. We are hoping she will be able to sleep on her decision and calm down.
"Chris Moyles has made it clear he's on standby if Sharon or Louis Walsh walk out at short notice. Simon Cowell is likely to take him up on that offer."