The single from Rice's debut album O has already been recorded by remaining finalists Amelia Lily, Little Mix, Marcus Collins and Misha Bryan, The Sun reports.
A source told the newspaper: "It's one of three songs being considered for release.
"The acts are still recording the different options but 'Cannonball' has a good shot."
They added: "After reworking Leonard Cohen's 'Hallelujah' for Alexandra Burke then Biffy [Clyro]'s track 'Many Of Horror' for Matt [Cardle] last year, it's the next step.
"It would be a safe bet to sell well in the run-up to Christmas."
'Cannonball' reached number 19 on its initial release in 2004 and re-entered the charts at number nine in September when it was performed during the X Factor auditions.
Earlier this month, N-Dubz star Fazer claimed that he would produce the X Factor winner's single.
"We really want to give a new flavour to the X Factor acts," he said. "It's going to suit me. I have rock influences, hip-hop, pop, whatever."
Previously, it was claimed that head judge Gary Barlow would pen the winner's song - making it the first original track for the purpose since Shayne Ward's 'That's My Goal' in 2005.
However, representatives for Barlow told Digital Spy at the time: "It has been talked about but it is way too early to tell what's going to happen with Gary and him writing any of the finalists' songs, including the winner."
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Watch the original Damien Rice 'Cannonball' music video below: