It is unclear whether Cole, 27, has accepted the offer; if she does, she will likely return to the United States when auditions for the Fox singing competition resume Thursday, THR reported.
It was never stated why Cole's services were terminated less than a month into the gig (auditions kicked off on May 8) but unnamed sources attributed the decision to Cole's heavy English accent.
After being let go, Cole was reportedly offered a chance to return to the British X Factor judges table. However, when the new team was announced on May 30, that list did not include Cole.
It is also unclear if she returns whether Scherzinger will return to co-hosting alongside Steve Jones.
X Factor creator Simon Cowell, former American Idol judge Paula Abdul and former record label chairman L.A. Reid also are set to serve as judges on the American X Factor, scheduled to debut on Fox this fall.