No. Scripture teaches that "it is appointed that men die once, and after this comes judgment" (Heb 9:27). There is absolutely no biblical evidence for reincarnation.
Sometimes people try to find biblical warrant for reincarnation in Christ's words about John the Baptist. In Matthew 17:12 Christ says, "I tell you Elijah has already come, and they did not recognize him." Matthew adds, "Then the disciples understood that he was speaking to them of John the Baptist" (Mt 17:13).
Is Jesus saying that John was the reincarnation of Elijah? No. Here's the simple reason. According to 2 Kings 2:9-18, Elijah was taken up bodily into heaven without seeing death. Thus, he wasn't a candidate for reincarnation because he was still in his original incarnation.
In Matthew 17:1-8, Moses and Elijah appear to Christ and some of his disciples at the Transfiguration. This occurs after John the Baptist has been executed by Herod Antipas. Why is it, then, that Moses and Elijah appear to Christ and his disciples, and not Moses and John the Baptist?
If Christ doesn't mean John the Baptist is the reincarnation of Elijah, what does he mean? Jesus is speaking figuratively in Matthew 17:12. He's comparing the prophetic ministry of John in the New Testament to that of Elijah in the Old. Similarly, Luke 1:17 says John "will go before him [the Lord] in the spirit and power of Elijah."
So there's no biblical basis for reincarnation. A person who is considering reincarnation is faced with a choice of believing other alleged sources of religious truth or believing the biblical witness. To accept the former in this instance is to reject the latter.