"Gilgul" is the Hebrew word for reincarnation, which comes from a verb meaning "turning in a circle."
The Old Testament prohibits talking to the dead. That in itself is not a direct endorsement of rebirth, but it indicates that the soul can be functioning quite well without a body, otherwise how could one talk with them? These souls must be somewhere after they die. Can they be in heaven or hell without first having judgment day? Or are they in heaven or hell because there is no "final" judgment day when all souls are judged and assigned permanently to either region? According to the Parable of the Prodigal Son, it would appear that there is no Judgment Day. Or, if there is a final judgment day, from what place are they now, before being judged, able to talk with living people?
These two scriptures assert the life of the soul after the body falls away:
Let no one be found among you who . . . is a medium or spiritist or who consults the dead. Deuteronomy. 18:10-11, NIV
Do not turn to mediums or seek out spiritists, for you will be defiled by them. I am the Lord your God. Leviticus 19:31, NIV
The next scripture asserts the existence of at least this soul prior to being placed in the womb.
The word of the Lord came to me, saying, "Before I formed you in the womb I knew you, before you were born I set you apart; I appointed you as a prophet to the nations." Jeremiah 1:4-5, NIV
In the Apocryphal book of Ecclesiasticus, at 41:8,9 of the KJV 1611 Edition, Solomon tells us in the following passage that if we do evil deeds in this life, we will have to reap the effects in our next incarnation. It is clear in this passage that Solomon believed in reincarnation. Having forsaken the law, these ungodly men will be reborn in another life to inherit their curse.
Woe be unto you, ungodly men, which have forsaken the law of the most high God: for if you increase it shall be to your destruction. And if you be born, you shall be born to a curse.
Flavius Josephus, who was born in 37 C.E. and died sometime after 93 C.E., is the most well know Jewish historian. He was a Pharisee, born Joseph ben Matthias. He served as a commander in Galilee in the Jewish revolt against Rome in 66 C.E. In his most famous book, "Antiquities of the Jews", written about 93 C.E., he describes the history of the Jews from the Creation onward with a full account of the Maccabees and the dynasty of Herod. Reincarnation is a prevalent theme throughout all of his works. He speaks of rebirth so matter-of-factly that one must assume it was an accepted way of thinking so common at the time that it didn't even merit any further explanation.
"The bodies of all men are, indeed, mortal, and are created out of corruptible matter; but the soul is ever immortal, and is a portion of the divinity that inhabits our bodies. . . . Do not you know, that those who depart out of this life according to the laws of nature . . . enjoy eternal fame; that their houses and posterity are sure; that their souls are pure and obedient, and obtain a most holy place in heaven, from whence, in the revolution of ages, they are again sent . . . into bodies; while the souls of those whose hands have acted madly against themselves are received by the darkest place in Hades?" ~The Works of Flavius Josephus, translated by William Wiston
Reincarnation was taught by three Jewish philosophers appearing just before or at the time of Christ. They were:
Philo of Alexandria, C. 20 B.C.E.- C. 50 C.E., the greatest Jewish philosopher and theologian of the Greco-Roman period whose writings have survived;
The Jewish sage Hillel, the great Chaldean teacher who was the leading Pharisee in Jerusalem during the late 1st century B.C.E. and early years of the present era;
The great Jewish sage Jehoshuah ben Pandira.
Today these three men are household names in Judaism and are highly respected by all orthodox Jews regardless of sect.
