I remember once seeing a Facebook post of an article claiming that the bible promotes reincarnation. This is one of many attempts at Christian syncretism with Dharmic religions. Another example is from those that say Jesus told people to “open their third eye chakra” based on what he said about the eye being the entry point of light in the body in Matt 6:22-23. This is clearly an allegory with light representing righteousness, and Jesus warning people about self-righteousness. The reincarnation idea is based on Matt 11:13-15 and Matt 17:10-13, in which Jesus says that John the Baptist is Elijah. There is a footnote in some translations to Malachi 4:5 here. Which says, “Look, I am sending you the prophet Elijah before the great and dreadful day of the Lord arrives.”
After the transfiguration (the appearance of Moses and Elijah with Jesus in glory form on the mountain), the apostles ask Jesus about the Elijah-John connection in Matt 17:10. It says (NLT) Then his disciples asked him, “Why do the teachers of religious law insist that Elijah must return before the Messiah comes?” 11 Jesus replied, “Elijah is indeed coming first to get everything ready. 12 But I tell you, Elijah has already come, but he wasn’t recognized, and they chose to abuse him. And in the same way, they will also make the Son of Man suffer.” 13 Then the disciples realized he was talking about John the Baptist. Other versions of this conversation can be found in Mark 9:11-13 and Luke 9:28-36.
The angel Gabriel actually tells Zechariah that his son John (the Baptist), will come in the “spirit and power of Elijah” in Luke 1:11-17. In John 1:19-28, John the Baptist is asked by servants of the Sanhedrin Council if he is the Messiah, Elijah, or the Prophet they are expecting and he says no to all three. We already know he [John the Baptist] was not the messiah, but Jesus himself in Matthew 17:11-13, and Gabriel in Luke 1:11-17 confirmed he is the Prophet announcing the Messiah’s ministry (Micah 4:5-6, Isa 40:3). John the Baptist had a “spirit” like Elijah, but he was not Elijah himself. So the answer is that he operates with the power of Elijah, not that he is Elijah reincarnated. This can’t be true anyway since Elijah never died. Remember Elijah got into a chariot of fire and was taken to heaven directly (2 Kings 2:7-11) so he didn’t actually die. This resembles how Enoch left earth since it’s implied he didn’t die either but was simply taken up by God (Gen 5:22). Some have suggested that the two prophets in Revelation 11:1-14 are Elijah and Enoch. These prophets cause plagues, call for repentance and preach against the Anti-Christ regime. They are killed, but then resurrected 3 days later. The idea is that since they never died in the first place they are brought back to earth for this moment in judging the world. Whether this is true or not is irrelevant to the point of this discussion. The main thing is Elijah didn’t die so he can’t be reincarnated, and even if he did die, the bible does not teach reincarnation. For more info on Reincarnation and Abrahamic theology click here.