Drake and 21 Savage have been sued by Vogue’s parent company Condé Nast over a fake magazine cover the duo used to promote their collaborative album Her Loss.

The rappers Drake and 21 Savage have been sued by Condé Nast, the publisher of Vogue magazine, for allegedly using the Vogue name without permission to promote their new album, Her Loss.

Condé Nast claimed the musicians’ promotional campaign, including to their more than 135 million social media followers, was built “entirely” on the unauthorised use of Vogue trademarks and false representations that they would appear on Vogue’s next cover, and with the “love and support” of the magazine’s longtime editor-in-chief, Anna Wintour.

“All of this is false. And none of it has been authorised by Conde Nast,” said a complaint filed on Monday night in a Manhattan federal court.

Condé Nast said the defendants also created a counterfeit issue of Vogue that was distributed in major metropolitan areas, accompanied by posters that mimicked Vogue’s layout.

It said the result was “unmistakable” confusion among the public, including media outlets that touted Drake and 21 Savage as Vogue’s “new cover stars”.

Larry Stein, a lawyer for the defendants, declined to make an immediate comment on Tuesday, having yet to review the complaint.

Condé Nast, also known as Advance Magazine Publishers Inc, is seeking at least $4m (£3.5m) in damages, or triple the defendants’ profits from their album and “counterfeit” magazine. It also wants punitive damages and an end to any trademark infringement.

Her Loss has received mixed critical reviews since its launch on 4 November.

Condé Nast said it had tried repeatedly since 31 October to resolve its differences with Drake, from Toronto, and 21 Savage, from Atlanta.

“[The] defendants’ flippant disregard for Condé Nast’s rights have left it with no choice but to commence this action,” it said.