Reincarnation of souls
Question

Who are the scholars who talk about reincarnation of souls and the life cycle?
Praise be to Allah.Firstly: What is meant by reincarnation of souls
What is meant by reincarnation of souls is that when a person dies, his body dies and his soul is released to dwell in another body, according to what he did of deeds in his previous life, and thus the soul begins a new cycle.
See: al-Mawsu‘ah al-Muyassarah fi’l-Adyan (2/728).
Secondly: Belief in reincarnation of souls constitutes disbelief (kufr)
Belief in reincarnation of souls constitutes disbelief (kufr). We have discussed this in the answer to question no. 14379 .
Al-Qadi ‘Iyad said: Likewise, we definitively state that he is a disbeliever who believes in the reincarnation of souls and says that they keep moving from one body to another for ever and ever, and that they will suffer torment or enjoy bliss in those bodies, according to whether they are good or evil."(Ash-Shifa’ 2/1067-1077).
Ad-Dardir al-Maliki said: A person becomes a disbeliever if he believes in reincarnation of souls. In other words, the one who says that whoever dies, his soul will move to another body like his, or one that is superior to it if the soul was obedient, or he will move to one that is inferior to it or similar to it if the soul was disobedient, is a disbeliever, because this is denying the resurrection.
End quote from ash-Sharh as-Saghir (6/147-148); Hashiyat ad-Dasuqi ‘ala ash-Sharh al-Kabir (4/269).
The Permanent Committee was asked: The philosophy professor told us that the soul moves from one human body to another. Is that true, and if it is true, how can it be the soul that is punished and brought to account, because if it moves then another person will be brought to account?
They replied:
What the philosophy professor told you, about the soul moving from one human body to another, is not true. The basic principle regarding that is the verse in which Allah, may He be exalted, says (interpretation of the meaning):
{And [mention] when your Lord took from the children of Adam - from their loins - their descendants and made them testify of themselves, [saying to them], “Am I not your Lord?” They said, “Yes, we have testified.” [This] - lest you should say on the day of Resurrection, “"Indeed, we were of this unaware.”} [Al-A‘raf 7:172].
The explanation of this verse is to be found in the report narrated by Malik in his Muwatta’, according to which ‘Umar ibn al-Khattab (may Allah be pleased with him) was asked about this verse (interpretation of the meaning):
{And [mention] when your Lord took from the children of Adam - from their loins - their descendants and made them testify of themselves, [saying to them], “Am I not your Lord?” They said, “Yes, we have testified.” [This] - lest you should say on the day of Resurrection, “"Indeed, we were of this unaware.”} [Al-A‘raf 7:172].
‘Umar (may Allah be pleased with him) said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) being asked about it, and the Messenger of Allah (blessings and peace of Allah be upon him) said: “Allah, may He be exalted, created Adam, then He touched his back with His right hand and brought forth some progeny from him, and He said: ‘I have created these for Paradise, and they will do the deeds of the people of Paradise.’ Then He touched his back and brought forth some progeny from him, and He said: ‘I have created these for the Fire and they will do the deeds of the people of the Fire…’”
Ibn ‘Abd al-Barr said: Regarding the meaning of this hadith, which was soundly narrated from the Prophet (blessings and peace of Allah be upon him) via many proven isnads from ‘Umar ibn al-Khattab, ‘Abdullah ibn Mas‘ood, ‘Ali ibn Abi Talib and Abu Hurayrah (may Allah be pleased with them all) and others,
Ahl as-Sunnah wa’l-Jama‘ah are unanimously agreed on that. They said that saying that the soul moves from one body to another is expressing the belief of those who believe in reincarnation, who are the worst of disbelievers and this belief of theirs is the falsest of falsehoods."(Fatawa al-Lajnah  2/435).
Thirdly: The scholars who talk about reincarnation of souls
Among those scholars who spoke about the reincarnation of souls are the following:
Al-Fasl fi’l-Milal wa’l-Ahwa’ wa’n-Nihal by Ibn Hazm (1/76 ff).
Athar al-Milal wa’n-Nihal al-Qadimah fi Ba‘d al-Firaq al-Muntasibah ila al-Islam, by Dr. ‘Abd al-Qaadir ibn Muhammad ‘Ata Sufi (69).
Ibn al-Qayyim also spoke about it in his books ar-Ruh (304) and Tariq al-Hijratayn (249-250).
For more information, please see: ar-Ruhiyyah al-Hadithah by Dr. Muhammad Muhammad Husayn (may Allah have mercy on him).