Praise be to Allah.
The skins from which footballs or basketballs are made can only be skins that have been tanned. Tanning means treating the skin with things that clean it and purify it, to remove any smell, decay or dampness. This is what usually happens. For the ruling on buying these things, it is essential to know the kind of animal from the skin of which it has been made; it can only be either an animal that it is permissible to eat or an animal that it is not permissible to eat.
If those things are made from the skin of an animal that may be eaten -- such as a cow -- then tanning makes its skin permissible. If it has been slaughtered according to shar‘i conditions, there is no need for tanning. But if it was slaughtered by people who go against the laws of Allah, may He be exalted, then it is maytah (dead meat), and the skin of an animal that may be eaten but died without being properly slaughtered can only become permissible by means of tanning.
If these things are made from the skin of an animal that cannot be eaten in the first place, such as a pig, then it is haraam to use its skin even if it has been tanned.
So with regard to animals whose meat may be eaten, slaughtering them in accordance with sharee‘ah makes their meat and skin permissible, and tanning makes their skin permissible if the animal died without being slaughtered in accordance with sharee‘ah (maytah or “dead meat”).
In the case of animals that cannot be eaten, slaughtering them in the shar‘i manner does not make their meat or skins permissible, and tanning does not make their skins permissible.
This is the view of al-Awza‘i, ‘Abd-Allah ibn al-Mubaarak, Abu Thawr and Ishaaq ibn Raahawayh; it is one view narrated from Imam Ahmad; and it is the view favoured by Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) according to one of his two opinions.
See: Sharh Muslim by al-Nawawi, 4/54, where he (may Allah have mercy on him) mentioned seven opinions on this matter; al-Furoo‘ by Ibn Muflih 1/102; Majmoo‘ al-Fataawa by Ibn Taymiyah 21/95.
This view was also favoured by a number of contemporary scholars.
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on him) said:
And it was said that the skin of an animal that died without being slaughtered correctly may be purified by means of tanning so long as it comes from an animal that may be made permissible by means of proper slaughter. But if it cannot be made permissible by means of proper slaughter, then it cannot be purified. This view is most likely to be correct. Based on that, the skin of a cat and any smaller animal cannot be purified by means of tanning.
The ruling is connected to whether the animal is regarded as taahir (pure) when it is alive. If it is taahir (pure), then it is permissible to use the skin of that animal if it died without being slaughtered properly, after tanning it once it has dried. But according to the madhhab it cannot be purified. According to the second opinion, it can be purified, and according to the third opinion, it becomes pure through tanning if it is from a dead animal that would be permissible (to eat) if slaughtered properly.
The most correct view is the third one, based on the fact that in some versions of the hadeeth it says: “Its tanning is its purification” (narrated by al-Nasaa’i, 4243; classed as saheeh by al-Haafiz ibn Hajar in al-Talkhees al-Habeer, 1/49). So the word purification was used, and it is well known that slaughter does not purify anything except that which it is permissible to eat. If you slaughtered a donkey and mentioned the name of Allah over it and shed its blood, it still could not be called halaal slaughter.
Based on this we say: the skin of an animal that it is haraam to eat, even if it is regarded as taahir when alive, cannot be purified by tanning.
The reason for that is that the animal that is taahir when it is alive was regarded as taahir because it is too difficult to avoid them completely, because the Prophet (blessings and peace of Allah be upon him) said: “They are among those who go around among you” -- referring to cats. This hadeeth was narrated by al-Tirmidhi (92), Abu Dawood (75), al-Nasaa’i (68) and Ibn Maajah (368), and was classed as saheeh by al-Bukhaari, al-Tirmidhi, al-Daaraqutni and al-‘Aqeeli, as stated by Ibn Hajar in al-Talkhees al-Habeer (1/41). This reason ceases to apply when the animal dies, whereupon it goes back to its original state, which is najaasah (impurity), and it cannot be purified by means of tanning.
So the correct view is that any animal that dies and may be eaten, its skin is purified by tanning. This is one of the two opinions of Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him); he has another opinion which is in accordance with the view of those who say that whatever was taahir when it was alive, its skin may be purified by tanning.
Al-Sharh al-Mumti‘, 1/74, 75
To sum up:
It is permissible for you to buy a football or basketball that is made from the skin of an animal whose meat may be eaten. Whatever was slaughtered in a way other than the shar‘i way or died of natural causes without being slaughtered at all, tanning its skin makes it taahir (pure) because of the hadeeth of Zayd ibn Thaabit (may Allah be pleased with him), from the Prophet (blessings and peace of Allah be upon him) who said: “Tanning the skin of the dead animal (that died without being slaughtered) is its purification.”
Narrated by Ibn Hibbaan (1290) and al-Daaraqutni (25); classed as saheeh by al-Albaani.
And Allah knows best.