All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Zakah disbursement channels are identified in Sharia and classified into 8 eligible categories. This is attested to in the verse in which Allah The most Exalted, says(What means):" Alms are for the poor and the needy, and those employed to administer the (funds); for those whose hearts have been (recently) reconciled (to Truth); for those in bondage and in debt; in the cause of God; and for the wayfarer: (thus is it) ordained by God, and God is full of knowledge and wisdom." [At-Tawbah/60].
Therefore, if the prisoners are poor, needy or indebted then they are eligible recipients, as indicated by the above verse. Further, prisoners are usually considered needy because they are unable to work or earn money; thus, this Zakah might save them along with their families.
In addition, Zakahis considered a valid act of worship once its conditions stipulated in Sharia are met, such as giving it to the eligible recipients and enabling them to act freely with it; therefore, paying their debts without seeking their consent is impermissible. In this regard, Al-Imam An-Nawawi (May Allah have mercy on his soul) said (What means):" Paying peoples' debts without their permission is impermissible but doing so with their consent is valid." [Rawdat At-Talibeen 1/258]. And Allah the Almighty knows best.