Jesus & John the Baptist were alive at the same time (tho John would've already been dead when they asked that - if we are to assume it's chronological). You can't really be a reincarnation of someone alive at the same as you. There's got to be another reason they were wondering if Jesus was John.
The idea that Jesus could have been John the Baptist does not fit w/ reincarnation, considering that they were contemporaries. And the idea that John the Baptist had been risen from the dead does not make sense either for the same reason.
Mar 6:14 King Herod heard of Him, for His name was spread publicly. He said, “John the Baptist has been raised from the dead, and therefore these miracles are at work in him.” 15 But others said, “He is Elijah.” And yet others said, “He is the Prophet, or like one of the prophets.” 16 But when Herod heard of it, he said, “It is John, whom I beheaded. He has been raised from the dead!”
Mat 14:1 At that time Herod the tetrarch heard of the fame of Jesus, 2 and said to his servants, “This is John the Baptist; he has risen from the dead. And therefore mighty works are at work in him.”
Luk 9:7 Now Herod the tetrarch heard of all that was done by Him. And he was perplexed, because it was said by some that John had risen from the dead, 8 and by some that Elijah had appeared, and by others that one of the old prophets had risen. 9 But Herod said, “I have beheaded John. But who is this of whom I hear such things?” And he tried to see Him.
I'm not exactly sure what Jesus's contemporaries thought about reincarnation in regards to John vs Elijah..
Jesus himself confirmed this to be the case (Mat 17:12-13).
I've always had it taught John had the 'spirit' of Elijah. He was metaphorically filling the prophecy.
This is the usual interpretation. But consider the healing of the man blind from birth:
Joh 9:1 As Jesus passed by, He saw a man blind from birth. 2 His disciples asked Him, “Rabbi, who sinned, this man or his parents, that he was born blind?” 3 Jesus answered, “Neither this man nor his parents sinned. But it happened so that the works of God might be displayed in him.
He was born blind because he had sinned? How could he have sinned before being born? The interesting observation is that the Lord did not refute the possibility, He only refuted that it happened that way.
That was my understanding as well since the Jews (at the time) did not believe in reincarnation. *
Belief in reincarnation was common in the Greco-Roman world and it appears to have been known (if not universally believed) in Israel.
*(it appears some modern Jews accept reincarnation but that is off topic)
This is true. Reincarnation is called "gilgul" in Hebrew.
Gilgul - Wikipedia