Matthew 11:13 For all the prophets and the law prophesied until John.
Matthew 11:14 And if ye will receive it, this is Elias, which was for to come.
Or
Matthew 16:13 When Jesus came into the coasts of Caesarea Philippi, he asked his disciples, saying, Whom do men say that I the Son of man am?
Matthew 16:14 And they said, Some say that thou art John the Baptist: some, Elias; and others, Jeremias, or one of the prophets.
https://en.wikipedia.org/wiki/Gilgul
On the other hand, the quotes from the NT I had always taken as types and forsahadowings, not as reincarnation. "This is Elias" means "He is the man who was prefigured by Elias." "He is fulfilling the role predicted by Elias."
In much the same way, when a new pope is elected, the choir sings "Tu Es Petrus" -- "You Are Peter" --- but it does not mean the pope is a reincarnation of Peter, it means he is the successor of Peter.