The 54-year-old judge is said to be furious about Dannii joining the panel for the new series, because she does not like sharing the spotlight with the Australian singer.
An ITV source told The People: "If this doesn't stop now it's going to be car-crash television once the live Saturday night shows start. Sharon is acting like a cow and poor Dannii doesn't know what's hit her.
"Sharon's nose has been well and truly put out of joint by the arrival of Dannii. In previous series she has been the star of the show."
The report claims that Dannii burst into tears following a row with Sharon at the Manchester auditions, and the pair refused to speak to each other at boot camp.
A spokesman for Sharon insisted: "Naturally there is rivalry as they want to win. But they have a mutual respect for each other. Any barbs are in jest."