The pair argue about 14-year-old Melanie McCabe from Dublin, who Simon believes is "100%" ready to progress in the competition.
However, Cheryl disagrees, saying: "I don't have to give her a shot because you are telling me I have to, I am going to make my own decision."
Shelley Morefield from Exeter also leaves the two judges at loggerheads. Cowell describes her rendition of 'Take Another Little Piece Of My Heart' as "everything I hate", while Cole calls it "gorgeous".
Last week's episode of the talent search drew more than 9 million viewers.
The X Factor airs at 7.30pm on Saturday on ITV1.