Dear I believe it is referring to Marquette...!!!

The eBay description says that Loome are 'known to have purchased a large portion of Tolkien's books from Catholic University, where he had donated his library'. The use of capitals, 'Catholic University', suggests Catholic University of America in Washington, D.C., which Daniel Grotta, at least, says (and we didn't entirely trust his account when writing the Companion and Guide, though maybe we should have cited it anyway) offered Tolkien a visiting professorship in 1946. We have no information about JRRT, or his estate, donating any of his books to Catholic University in Washington (or any other 'Catholic University', e.g. in Nijmegen). Nor do we know of any of his books going by donation to Marquette.

Looking at other sources, such as this thread on the present site in 2008, one finds references to Loome purchasing books merely from 'a Catholic university', which opens the field and makes no mention of a donation - and any university with a Catholic foundation may have purchased books from Tolkien when he downsized his library or when his books were sold after his death.

And beyond this, we wish that there was a larger and clearer picture of the signature on the item in question, because it doesn't strike us immediately as Tolkien's handwriting.

Note to Urulókë: don't bother e-mailing Matt at Marquette, he left to head the library of the Wisconsin Historical Society. The new director of the Archives and Special Collections at Marquette is Amy Cooper Cary.