Matthew 17:10-13 is the usual proof-text for reincarnation. It centers on Elijah and JBap. Dale already posted the normal proof-text hunt against reincarnation. With how malleable scripture is in the face of errors, contradictions and competing theology to many Christians, I don’t think either one necessarily precludes the other.
10 The disciples asked him, “Why then do the teachers of the law say that Elijah must come first?”
11 Jesus replied, “To be sure, Elijah comes and will restore all things.(A) 12 But I tell you, Elijah has already come,(B) and they did not recognize him, but have done to him everything they wished.(C) In the same way the Son of Man is going to suffer(D) at their hands.” 13 Then the disciples understood that he was talking to them about John the Baptist
I find the notion of purgatory reasonable. Never been convinced reincarnation is a thing but I know some point to “memories of past lives.”
Edited to add, I suppose it can be said from Matthew’s perspective Elijah never official “died” even though he was gone from the universe.