When a woman is making up for the fasts that she has missed in Ramadan, is it permissible for her to break her fast if her husband wants to have sexual intercourse with her or not, knowing that she sought his permission before fasting? What should she do if he threatens her with divorce?

All perfect praise be to Allah, The Lord of the Worlds. I testify that none is worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.

If one starts an obligatory fast, such as making up a missed Ramadan fast, offering expiation, or the like, it is not permissible for him to break it and it is obligatory for him/her to complete it. The author of Ash-Sharh Al-Kabeer said:

It is not permissible for a person who started an obligatory fast—such as making up obligatory fasts, making up for Ramadan, fulfilling a specific or unspecific vow, or fasting for expiation—to break that fast. That is because an obligatory act of worship becomes binding by starting it and what is not obligatory becomes obligatory once the person starts it, and thus it becomes like the obligatory one. There is no disagreement among scholars regarding this issue, and praise be to Allah.

This is proved by what Abu Daawood and Sa‘eed ibn Mansoor and Al-Athram reported from Umm Haani’ who said:

A vessel (of drink) was brought to the Prophet, sallallaahu ‘alayhi wa sallam. He gave it to me and I drank of it. Then I said, “O Messenger of Allah, I broke my fast when I was fasting.” He, sallallaahu ‘alayhi wa sallam, said to her: “Were you making up for something?” She (I) said: “No.” He said: “Then it does not matter if you were observing a voluntary fast.”

Accordingly, you should know that it is not permissible for you to break the fast intended for making up missed fasts of Ramadan upon the will of the husband. However, you have to tell him and seek his permission before starting the fast. Then, if he gives you permission to fast and after you start fasting, he wants to have sexual intercourse with you, then it is not permissible for you to obey him, even if he threatened you with divorce. Allah, The Almighty, said (what means): {...And whoever fears Allah—He will make for him a way out.} [Quran 65: 2]

Also, there is no obedience to any human being if it involves disobedience to The Creator.

Allah Knows best.