The slime's earliest codification in games is from the original 1974 edition of Dungeons and Dragons Monsters and Treasure, where it is listed as a hazard rather than a monster:

A non-mobile hazard, Green Slime can be killed by fire or cold, but it is not affected by lightening bolts or striking by weapons. It eats away wood and metal but not stone. Green Slime sticks to flesh and penetrates it in one turn, thereafter turning the flesh into Green Slime. Green Slime cannot be scraped off, so when it contacts something the item must be discarded or excised in some way. A Cure Disease spell will also serve to kill and remove Green Slime, even when it is contact with flesh.

Also in that edition are the monsters Ochre Jelly, Black Pudding, Gray Ooze, and Yellow Mold. The concept of a sentient, malevolent ooze goes back to the monster serials of the 1930s-60s, which influenced Gary Gygax heavily. "The Blob" comes to mind, but there were several others. They, in turn, hearken back to Lovecraft's At the Mountains of Madness (1931), where we find the monstrous Shoggoths. To my knowledge, there is no earlier clear attestation of such a creature with those general properties.

In video games, I believe the first appearance of a recognizable slime enemy is from Wizardry: Proving Grounds of the Mad Overlord (1982), where we find the Bubbly Slime. This game and its successors heavily influenced the Japanese developers responsible for the well-known console RPGs of the mid-80s, especially Dragon Quest and its familiar Suraimu creature, the most basic enemy of the series.

u/Alkibiades415 has already mentioned Lovecraft but I don't think that the influence of Clark Ashton Smith should be discounted either. On the website rpg.net Mike Mornard, the youngest of Gary Gygax's original D&D group used to post extensively under the handle "Old Geezer." According to him Gygax's favorite author was Lovecraft but he and Rob Kuntz (another very early member of Gygax's original play group) preferred Clark Ashton Smith.

Among Clark Ashton Smith's many many short stories, Mornard specifically mentions the story The Seven Geases (www.eldritchdark.com/writings/short-stories/192/the-seven-geases) repeatedly in his rpg.net postings.

That story includes the following passage:

"The dusk about him thickened with hot, evil steam that left an oozy deposit on his armor and bare face and hands. With every breath he inhaled an odor noisome beyond imagining. He stumbled and slipped on the crawling foulnesses underfoot. Then, in that reeky twilight, he saw the pausing of Raphtontis; and below the demoniac bird he descried a sort of pool with a margin of mud that was marled with obscene offal; and in the pool a grayish, horrid mass that nearly choked it from rim to rim.

Here, it seemed, was the ultimate source of all miscreation and abomination. For the gray mass quobbed and quivered, and swelled perpetually; and from it, in manifold fission, were spawned the anatomies that crept away on every side through the grotto. There were things like bodiless legs or arms that flailed in the slime, or heads that rolled, or floundering bellies with fishes' fins; and all manner of things malformed and monstrous, that grew in size as they departed from the neigbborhood of Abhoth. And those that swam not swiftly ashore when they fell into the pool from Abhoth, were devoured by mouths that gaped in the parent bulk."

Mike Mornard's postings about Smith can be easily found by searching rpg.net. Here's one of them:

"One of Gary's favorite stories, and Rob Kuntz' very favorite story, is "The Seven Geases" by Clark Ashton Smith.
'Nuff said."

https://forum.rpg.net/index.php?threads/early-editions-d-d-is-a-horror-game.449951/post-10284522