Sheikh Muhammad al-Sarî`, professor at al-Imâm University in Riyadh
Al-Salâm `Alaykum wa Rahmah Allah wa Barakâtuh.
You may not sterilize the female cat. It would be better if you can offer it some means of accommodation together with its litter or give it to someone who is willing to do so.
Likewise, you may not neuter the male cat.
Scholars prohibit sterilization categorically, even for animals, except in cases where there is a real necessity or tangible benefit from doing so. Keeping a pet is a mere luxury.
Al-Qurtubî states: “It is prohibited to castrate an animal except for a tangible benefit like improving the quality of meat or preventing harm.”
Al-Nawawî writes: “It is categorically forbidden to castrate animals that cannot be used for food. As for animals that can be used for food, it is permissible to castrate the juveniles but not the adults.”
And Allah knows best.
Source: Islam Today