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 is His slave and Messenger.
If the bank is run by a legal (Islamic) committee that arranges its dealings and by an Islamic body that watches and organizes its transactions, then it is an Islamic bank which can be dealt with. But if it does not have such qualities, then it is not allowed to deal with it for it is not an Islamic bank though it may be called so. This is as far as the bank itself is concerned. As concerns Riba (interest and/or usury), it could be got in return for unreal selling process where no actual sale is conducted. This happens when the bank does not really buy a certain property nor does it possess it as its own belonging, and then it sells it to the buyer. If this is the case, then the profit will be considered Riba. So, dealing with such kind of banks is prohibited. However, if the profit is as an income gained in return for a transaction conducted between the bank and the client after the bank really buys the property and possesses it and then sells it to the client, then this kind of sale is called Murabaha (Selling at a Profit). This kind of selling is lawful and actually it is the common practice of the Islamic banks.
Allaah Knows best.