All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
If the money of the shareholders of this bank is totally unlawful, it is not permissible to deal with this bank. But, if their money is mixed, i.e. combining both lawful and unlawful money, it is permissible to deal with this bank on condition that its transactions are regulated by the boundaries of the Sharee‘ah.
In fact, it is not enough that the bank claims to be Islamic. For further information about the work regulations of the Islamic banks, see Fataawa 86125 and 83941.
Allaah Knows best.