Is saving money and placing it in the names of daughters in an Islamic bank permissible or forbidden, knowing that they have no income other than this saved money?

If the money is deposited in the name of the two daughters and is not an immediate gift, it is considered an inheritance after death and is to be divided among all heirs. If the transfer of ownership is contingent upon death, it is a bequest to an heir, which is not valid unless approved by the other heirs. However, if you remove the money from your ownership and gift it to them during your lifetime and in good health, depositing it in their names, and you no longer have control over it except in a guardian capacity, then there is no harm in it, and it becomes their property. Witnessing this is sufficient, and physical possession is not required if they are minors.