In the December 2018 issue of Mallorn: The Journal of the Tolkien Society, Dale Nelson considers tentative connections between H. P. Lovecraft and the Oxford Inklings.
Devotees of fantastic fiction have wondered if the Ink-lings knew the work of the Lovecraft circle, and vice versa. Several points of likely or certain awareness may be sum-marized as follows:
 By sometime late in his life, Clark Ashton Smith, short story writer, poet, and artist, had read The Hobbit and some of The Fellowship of the Ring, according to a posting by “calonlan” on 30 Nov. 2011, in an Eldritch Dark discussion thread. “Calonlan” appears to have known CAS personally.
 Lovecraft himself had read more than one of Charles Williams’s spiritual thrillers. Their orthodoxy spoiled them for HPL. He wrote:
“Essentially, they are not horror literature at all, but philosophical allegory in fictional form. Direct reproduction of the texture of life & the substance of moods is not the author’s object. He is trying to illustrate human nature through symbols & turns of idea which possess significance for those taking a traditional or orthodox view of man’s cosmic bearings. There is no true attempt to express the indefinable feelings experienced by man in confronting the unknown . . . To get a full-sized kick from this stuff one must take seriously the orthodox view of cosmic organisation—which is rather impossible today.” (as quoted in S. T. Joshi, I Am Providence, page 878; I’m indebted to a 21 April 2016 posting by John Rateliff on his Sacnoth’s Scriptorium blog for this reference)
Lovecraft could not have read Descent into Hell and All Hallows’ Eve, which contain perhaps the most “Lovecraft-ian” sequences in Williams’s seven novels.
 Lewis almost certainly not only read, but was influ-enced by, a story by Lovecraft correspondent and Arkham House co-founder Donald Wandrei. On one of the last pages in his short novel The Great Divorce, Lewis acknowl-edges his indebtedness to an American science fiction story, the title and author of which he has forgotten. This appears to be “Colossus,” which appeared in the January 1934 issue of Astounding. Wandrei’s story plays with the idea of our universe being of subatomic tininess as com-pared to a super-universe; the hero journeys from the one to the other. Lewis’s novel involves a bus trip from hell to heaven. In the fiction, “‘All Hell is smaller than one pebble of your earthly world; but it is smaller than one atom of this world, the Real World.’”
 Tolkien evidently read a 1963 paperback anthology called Swords and Sorcery, edited by L. Sprague de Camp, who gave him a copy. The anthology contains Lovecraft’s tale in the manner of Lord Dunsany, “The Doom That Came to Sarnath,” Smith’s “The Testament of Athammaus,” and Howard’s Conan story “Shadows in the Moonlight.” According to de Camp, who visited Tolkien in 1967, Tolkien liked the Conan story. Tolkien’s own copy of de Camp’s anthology was offered for bids on ebay a few years ago. http://www.tolkienlibrary.com/tolkien-book-store/000971.htm
 It is reasonably likely that Lewis read Lovecraft’s At the Mountains of Madness and “The Shadow Out of Time” in Astounding. His reading of American pulp magazines is certain. Below, I’ll say something about possible influ-ence of Mountains on Lewis’s Out of the Silent Planet and of “Shadow” on Lewis’s Dark Tower fragment.
All that doesn’t come to a lot, but it’s more than might have been expected.
Tolkien reading Conan later in life is obviously the most satisfying of these nebulous associations. Nelson addresses Lovecraft’s partially formed mythopoeic imagination in greater detail. The essay can be read at the link above.