"You have exactly 48 hours from now to release a statement apologizing to @AzealiaBanks and the protesters in NYC," reads a message posted Saturday from the now-defunct account @TheAnonMessage, and captured by Vulture.com.
"You are guilty of misappropriating black culture, insulting peaceful protesters and making light of Eric Garner's death," read another message.
"There's an X rated tape of you, we bet you'd certainly don't want public, are we right? @IGGYAZALEA....and this is nothing. Comply or else."
Twitter has since suspended the account.
Azalea has previously denied the existence of a sex tape, and has said: "Anyone who releases or attempts to make profit off someone else's intimate moments against their will is a sex offender."
It's not immediately clear what Azalea might have done to "make light' of the death of Garner, an unarmed black man, and the nationwide protest movement against police brutality.
Threats refer to ongoing feud
The threat appears to relate back to a public feud between Banks and Azalea that heated up when Banks slammed Azalea in an outspoken radio interview last week in New York in which she complained about the cultural appropriation of black culture by white hip hop artists.
"That Iggy Azalea s--t isn't better than any f--king black girl that's rapping today," Banks said to New York's Hot 97.
The 23-year-old also suggested that Azalea didn't deserve her recent Grammy nomination for best rap album because her music "is not excellent" and doesn't deserve to compete "against black culture."
Azalea took issue with Bank's comments, responding on Twitter, "There are many black artists succeeding in all genres. The reason you haven't is because of your p--s poor attitude."
Neither Azalea nor Banks has publicly commented on the Anonymous threat.