Tolkien did own an anthology of stories that included a story by Lovecraft. Seems likely he read it. Whether it influenced his writing is another matter though :).

Fantastic (magazine) November 1976 [Discussion Forums - Fanzines and other Periodicals]

Discussion of whether he was influenced here, which I found with google search (I'm not the person asking the question). Tolkien Topics: Reading Room: Tolkien & Lovecraft

More background here:

In July 1964, de Camp sent J.R.R. Tolkien a copy of his anthology Swords & Sorcery.[1] Tolkien sent a letter to de Camp in August 1964, expressing his opinion about the stories.[2] In February 1967, de Camp and Alan E. Nourse visited Tolkien for an interview:[3][4]

"[Tolkien] said he found [the anthology] interesting but did not much like the stories in it [...] We sat in the garage for a couple of hours, smoking pipes, drinking beer, and talking about a variety of things. Practically anything in English literature, from Beowulf down, Tolkien had read and could talk intelligently about. He indicated that he 'rather liked' Howard's Conan stories."― Literary Swordsmen and Sorcerers, pp. 243-4

from: L. Sprague de Camp

So - at least in 1964 he almost certainly read one of his stories, "The Doom that came to Sarnath" though he probably didn't much like it, at least, didn't much like the anthology it was in, though he didn't particularly mention it, so who knows for sure?

That's after he wrote the Hobbit and the Lord of the Rings. Of course that doesn't say anything either way about whether he read any of his stories before then.

It's a bit of a slender link, but best I can find :).