Drake, 21 Savage Sued by Vogue Over Fake Magazine Cover Promotions

Drake and 21 Savage's hot new album, "Her Loss" is gonna be their loss in court ... if Vogue gets its way -- the magazine's suing over fake covers the rappers used to hype their record.

According to legal docs, obtained by TMZ Hip Hop, Drake and 21 have been promoting "Her Loss" by posing on what certainly looks like a legit Vogue cover, leading fans to believe the company would drop the special edition soon -- some media outlets even reported on the issue.

Condé Nast, which owns Vogue, says that isn't the case ... despite Drake giving the brand and editor-in-chief Anna Wintour a shout-out online for getting a blessing.

Condé Nast claims Vogue and Anna "have not endorsed ["Her Loss"] in any way" ... and have asked multiple times for all promos using the Vogue name to be taken down, but nothing's been done.

The docs say the rappers falsely promoted the Vogue collab on social media, and put up posters of the "cover" in major cities.

Condé Nast also claims Drake and 21's teams sent out an email blast about "celebrating Drake's Vogue cover" ... saying street teams across the country would even be handing out copies.

People on the streets allegedly got their hands on the false mags, according to the docs ... and many claim it's a direct replica of Vogue -- even including a fake pic of Anna posing with Drake. Wow!!! Can't say they didn't go all out.

Condé Nast wants a judge to shut down the use of the Vogue mark immediately ... and it wants at least $4M in damages.

We've reached out to Drake and 21 Savage for comment ... so far, no word back.