Ran across a few people on reddit, and in my own life, saying that this was a thing. My own research didn't find too much in hard references, as I'm unfamiliar with the historical researching side of things as opposed to the scientific databases. Any insights would be great! I'd love to learn more.
I've discussed this in a much older thread of a post of mine, on a related topic (you need to follow the discussion through to where it turns to resurrection specifics).
The answer is that there's no evidence of a belief in reincarnation among early Christians. It's not a totally uncommon thing people like to say, and I'd be happy to discuss further any particular arguments put forward, or that you've encountered (and happy to revisit points from that linked thread). But there's simply zero evidence of this belief across the early church/late antique period.