Reincarnation is the teaching that after death, the spirit goes to a holding place and then enters a new human body during conception -- to be born again or to experience another physical birth. The memories of their past life are retained.
This is somewhat similar to Christ's birth. He retained his memories of his past life and experienced a physical birth -- the Word became flesh. But Christ was from spiritual to physical. Change this to physical to physical, it would be a reincarnation.
I know that reincarnation is against the Bible according to some Christian apologetics sites.
However, there are some verses which hints reincarnation.
"And as he was passing by, he saw a man blind from birth. And his disciples asked him, 'Rabbi, who has sinned, this man or his parents, that he should be born blind?" Jesus answered, 'Neither has this man sinned, nor his parents, but the works of God were to be made manifest in him.'" (John 9:1)
Why did the disciple ask Christ this question? Was there a previous life this blind man had that he should pay for his sins when he is born again in this world, that is, reincarnated? The disciple was not even refuted by Christ.
Then there is also a passage about John the Baptist as Elijah, if we are willing to accept it.
Is there anything impossible for the Lord to do? What if the Lord allows for reincarnation for certain individuals? He did allow Lazarus to be resurrected and be dead again the second time. He did allow Himself, the Creator of all things to become flesh and blood.
I think with the Lord anything is possible. But, we are so afraid of accepting things that we think have no Biblical basis, therefore should be dismissed altogether and simply stick to a strict law of, "we die only once, after that the judgment." Even the Jews cannot accept the fact that the Lord can become a human being. It's so alien to them. They are better off believing what the Torah said that the Lord is not a mortal.
However, just like Lazurus, these holy people were raised to life after Jesus' resurrection:
Mat 27 51 At that moment the curtain of the temple was torn in two from top to bottom. The earth shook, the rocks split
52 and the tombs broke open. The bodies of many holy people who had died were raised to life.
53 They came out of the tombs after Jesus’ resurrection and went into the holy city and appeared to many people.
Of course, these people died for the second time and will be resurrected again - twice died and twice resurrected.
So, the law of "we die only once, after that the judgment" doesn't strictly apply to everyone. We shouldn't really put a limitation to what the Lord can do just because the Bible does not elaborate much about it, only leaving some breadcrumbs to figure it out.