Short answer
It depends. Freezing a woman's own eggs for a medical reason, such as before cancer treatment, is allowed by many scholars, provided the eggs are later fertilised only with her husband's sperm during a valid marriage and placed in her own womb. Freezing eggs without a medical reason, to delay marriage or motherhood, is treated with more caution.
There is no classical text; Hanafi muftis apply the IVF conditions. Freezing one's own eggs, or a couple's embryos, is permissible for a medical need with strict safeguards. They may be used only by the same couple while the marriage lasts, never after divorce or the husband's death, and never by anyone else.
The four schools agree on this.
Explanation
Egg freezing raises two questions: is it allowed to collect and store the eggs, and how may they be used later? The eggs are part of the woman herself, so storing them is not in itself a problem. The rules for using them are the same as for IVF: only with her husband's sperm, only while they are married, and only in her own womb (see is IVF haram).
Medical freezing. When a woman faces chemotherapy, ovarian surgery or a condition that will end her fertility early, freezing her eggs is a form of treatment, and treatment is encouraged. The uncovering of the awrah it needs is allowed, with a female specialist preferred.
Freezing without a medical reason. Contemporary scholars differ. Some allow it with the same conditions. Others discourage it because it involves hormone treatment and uncovering the awrah without a present medical need, carries the risk of mix-ups in storage, and can encourage delaying the marriage that the Sunnah urges. If you are considering it, put your situation to a mufti.
Whichever case applies, give the clinic clear written instructions: the eggs or embryos are never to be donated, and are to be discarded if you die, if the marriage ends, or if you no longer need them. Frozen embryos may not be used after a divorce or the husband's death, because the marriage they belong to has ended. The four schools have no classical ruling here, and contemporary scholars of all schools apply these same IVF conditions.
Ask a mufti about your own case. This answer explains the general ruling. When the details of your situation matter, as they do here, ask a qualified mufti and give them the full facts. Where to ask a scholar.
The evidence
Quran, Surah Ash-Shura 42:49
لِّلَّهِ مُلْكُ ٱلسَّمَٰوَٰتِ وَٱلْأَرْضِ ۚ يَخْلُقُ مَا يَشَآءُ ۚ يَهَبُ لِمَن يَشَآءُ إِنَٰثًۭا وَيَهَبُ لِمَن يَشَآءُ ٱلذُّكُورَ ﴿٤٩﴾
To Allah belongs the dominion of the heavens and the earth; He creates what he wills. He gives to whom He wills female [children], and He gives to whom He wills males.
Children are a gift from Allah, and permitted means of preserving fertility are part of seeking it from Him.
Sahih Muslim 2204
Sahih... لكل داء دواء فإذا أصيب دواء الداء برأ بإذن الله ...
"Every disease has a remedy, and when the right remedy is found, the disease is cured by Allah's permission."
Preserving fertility before a harmful treatment is a form of medical care.
Sunan Abi Dawud 2158
Graded hasan... لا يحل لامرئ يؤمن بالله واليوم الآخر أن يسقي ماءه زرع غيره ...
"It is not lawful for a man who believes in Allah and the Last Day to water another man's crop with his own water."
Stored eggs may only ever be fertilised by the woman's own husband.
Where the ruling is recorded
- Resolutions of the Islamic Fiqh Academy (OIC), Resolution on test-tube babies. Sets the conditions of husband, wife, her womb and a living marriage.
- Radd al-Muhtar (Ibn Abidin), Kitab al-Hazr wa al-Ibahah, Fasl fi al-Nazar wa al-Mass. Looking at the awrah is allowed for treatment, and only as far as needed.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.