Carey, who was officially named as a new judge in July, was previously rumoured to be unhappy about speculation that Minaj could join the show.
Minaj was confirmed to be on the panel yesterday, along with Keith Urban and continuing judge Randy Jackson.
However, she has already clashed with Carey during taping, TMZ claims.
The website alleges that during auditions in New York City yesterday (September 16), Carey would interrupt repeatedly as Minaj tried to offer critiques to contestants. In turn, Minaj reportedly spoke loudly over Carey.
"These girls just don't like each other," a source claimed.
Minaj later told paparazzi outside the American Idol auditions that she had a "great" time on her first day, saying: "I can't complain, it was great. I'm just tired and a little bit hungry!"
American Idol is expected to return to Fox early next year.