I think I found the source of this theory.
Google scholar has turned up this 1930 book: From Orpheus to Paul a History of Orphism. One recent publication that cites it is a 2008 thesis of Stian Sundell Torjussen from the University of TROMSØ. I'll quote a passage from p.40 there:
Orphism was thus seen as taking a step beyond Greek religion, forming
the vanguard of the Greeks evolutionary path towards monotheism. The
first steps on this road were, as Macchioro argued, taken by the
philosophers, especially Plato, who, deeply influenced by Orphic
thought and doctrines, developed a system of thought which ultimately
laid the foundations for Christianity.
However, Torjussen goes on to review work by later scholars, Willamowitz and Linforth in particular, who, according to him, demolished some of the more far-fetched claims of the Macchioro school (pp. 41-44).
It's actually quite fascinating reading, Torjussen does a very good job summarizing the debates.
I hope it answers your question!
Although I am not an expert by any means and can't even begin to evaluate at present the evidence of the allegedly Orphic tablets which seems to play a large role in this whole business. However, I'd like to throw in my own two cents: that early Christianity was influenced by Neoplatonism is well-known even to laymen like me, but I've always assumed that the influence comes from a stage a few centuries later (say, Origen). The attempt to backdate this to Paul is, in my eyes, a bit flimsy, because Paul was a Jew and could hardly be directly familiar with Greek rituals and mysteries before his conversion. So to posit that Paul was influenced by Orphism, one has to postulate some intermediate link, such as the Essenes - which lifts the whole thing to the rarefied realm of almost-unverifiable conjecture. But that's just my two cents, they could be total piffle, read Torjussen anyway!