The simple answer is no. When proponents of reincarnation allege that certain biblical texts teach the soul's preexistence or reincarnation, they are approaching those texts superficially and their interpretations dissolve under further scrutiny.
Reincarnation (Hinduism) or rebirth (Buddhism) is integral to Eastern philosophy. In the Bhagavad Gita, Krishna talks of having "passed through many births." And what we reap in this life (karma) comes from what we've sown in past lives. Biblical, theological, and philosophical reasons, however, undermine reincarnation.
If one acknowledges the Bible's authority and storyline, one will readily recognize the Eastern doctrine of reincarnation as unacceptable. Many claiming that reincarnation appears in the Bible would go on believing in reincarnation anyway, with or without biblical support. They read reincarnation into isolated verses (e.g., statements about being "born again" in Jn 3) without respecting the biblical context or the worldview of the author. In doing so, they do not respect the biblical text as they would want their own Eastern texts respected. (What if we read bodily resurrection into their texts?)
Each of us must die and then be judged by God (Heb 9:27). When God told Jeremiah He knew him before he was in his mother's womb (Jr 1:5), this doesn't demonstrate preexistence or reincarnation; it only indicates God's foreknowledge and sovereignty. Notice Jeremiah did not say, "Before I was in my mother's womb, I knew You, God." That would make a persuasive case for preexistence! Also, the disciples' questioning whether the man born blind sinned before birth (Jn 9:2) does not express reincarnation but rather reflects the rabbinic belief that a fetus could sin while in his mother's womb (cp. Genesis Rabbah 63.6).
Furthermore, the historically supportable event of Jesus' bodily resurrection undercuts reincarnation. The biblical view of the afterlife is radically different from that of Eastern philosophies. True immortality is not the eradication or "snuffing out" (moksha) of the self nor its absorption with the One, Brahman, like a drop in an ocean. To receive immortality is to receive an immortal, imperishable physical body (1 Co 15:53-34). It is a spiritual body (that is, one supernaturally animated by the Holy Spirit) rather than a natural body (animated by a human soul). Immortality means being forever in union with God and living in God's presence with this new body in the new heavens and new earth-without losing individual identity.
Theologically, God's grace and forgiveness undercut karma. We need not bear the heavy weight of guilt and shame because Jesus Christ has absorbed all that for us. And if reincarnation is true, why help the underprivileged? Aren't they getting what they deserve-their karma?
Despite "evidence" for reincarnation, arguments for a person having lived previous lives could be explained by demonic activity (see Ac 16:16-18). A person having access to information about another's previous life does not imply that this was his own life. A psychic may purport to have knowledge of a crime, but this doesn't mean he committed it!
Philosophical problems with reincarnation are many. (1) Those "remembering" past lives tend to be clustered in the East (where reincarnation is taught), not throughout the world (as we'd expect). (2) If we forget our past lives, what purpose does reincarnation serve for self-improvement? (3) Assuming reincarnation (with an infinite past series of rebirths), then we've all had plenty of time to reach perfection. Why haven't we? (4) Reincarnation doesn't solve the problem of evil, as some claim, but only infinitely postpones it (and in some Eastern schools, evil is just an illusion anyway). (5) Reincarnation makes incoherent the Eastern idea of monism, which says that everything is one without distinction, by presupposing distinctions between (a) individual souls, (b) the karmas of individual souls not having yet reached enlightenment, (c) the enlightened and unenlightened, and (d) individual souls and the One (ultimate reality).
Extracted from the Apologetics Study Bible.