The rapper just dropped the music vid for "Temporary" -- and in one scene Eminem and Hailie are sitting on a set of stairs on a porch when she hands him a custom Detroit Lions jersey ... that says "Grandpa" on the back.

Hailie also gives her dad a sonogram of the fetus ... and he holds it up toward the camera with a surprised look on his face.

A source with knowledge of the situation confirms Hailie is indeed pregnant ... and this isn't just some stunt for the music video.

Unclear how far along Hailie is, and ditto for the sex of the baby ... but Hailie just got married back in May to Evan McClintock.

The music video features a ton of footage of Hailie growing up over the years ... and Eminem even includes part of her wedding, where he's super emotional.

"Temporary" is from Em's latest album, "The Death Of Slim Shady (Coup De Grâce)" ... and Slim Shady's name-dropped Hailie in countless songs throughout his career.

Now, she's about to be a mother ... and Eminem's about to become a grandfather for the first time.