The release of the video, which TMZ reported was recorded during an audition of "Idol" hopefuls in Charlotte, North Carolina, follows rumors that Minaj and Carey had been feuding.
"Idol" broadcaster Fox is looking to the new line-up of judges, which includes country star Keith Urban, to help revitalize a show that has seen falling ratings in recent years.
Minaj, 29, a rap star known for her outlandish outfits, is shown in the TMZ video yelling and cursing. It is not clear from the clip, which is less than a minute long, what might have led to the outburst or who shot the video.
"I told them, I'm not (expletive) putting up with her (expletive) highness over there," Minaj is seen saying at one point in the video, as she turns toward Carey.
Carey can be heard in the video speaking in the direction of Minaj, but her words are less audible.
Urban, who is seated between the two women pop stars, at one point raises his arms, and fourth judge Randy Jackson - whose tenure on the show dates from its 2002 debut - can be heard saying "Hold up" in an apparent effort to defuse the conflict.
A Fox spokeswoman declined to comment on the video.
Minaj, Urban and Carey were brought onto the "Idol" judging panel in recent months after Aerosmith frontman Steven Tyler and singer-actress Jennifer Lopez each left the show following a two-year stint.
"American Idol" has been a ratings juggernaut since it debuted on Fox a decade ago, but audiences are now only about half the more than 30 million who watched regularly in the show's 2005 and 2006 heyday.
Minaj has emerged as the top female rapper in a hip-hop world dominated by male artists. She burst onto the music scene in 2010 and has become famous for her numerous alter-egos and her multiple hit records, including "Starships."
A representative for Minaj could not immediately be reached for comment.
"Idol" fans were divided about the selection of Minaj for the judging panel. Her appeal to young audiences is seen as potentially key to helping "Idol" regain youthful viewers, but she has also proved controversial. In February, she appeared on the Grammy awards red carpet with a fake pope.
