The bank is asking him to open two accounts, one for saving, on the grounds that it will protect his money
Question

                I have an account in an interest-based bank, but without interest. When the money in my account reached a specific amount, the clerk in charge talked to me and asked me to open another account to protect my money. Thus I will have two accounts; one of them is to deal with by the credit card and cheques, and I do not take any interest on this one. While the other is to save most of my money in it, I can take from this account if I go personally to the bank, and I will take interest on this one.  

They force you to open another account and to take interest on it, so that you protect your main account. Is this permissible, knowing that having an account in an interest-based bank is a necessity in the countries of kufr?.
Praise be to Allah.
    It is not permissible to put money in riba-based banks except 
    when there is a need for that and when there are no Islamic banks. In that 
    case one should only use current accounts that bear no interest, as a case 
    of committing the lesser of two evils. Otherwise opening a current account 
    in a riba-based bank is also haraam, but the prohibition is less severe than 
    in the case of a savings account. The reason why it is haraam is that the 
    bank will make use of this money and use it to help in its haraam 
    transactions such as riba etc. 

    But if a person fears for his money, then it is permissible 
    for him to put money in these banks. 

    It says in Fataawa al-Lajnah al-Daa’imah (13/346): 

    It is not permissible to put cash etc in riba-based banks and 
    other riba-based institutions, whether that is with or without interest, 
    because that is a kind of cooperating in sin and transgression, and Allaah 
    says (interpretation of the meaning): “do not help one another in sin and 
    transgression” [al-Maa'idah 5:2]. But if he fears that his wealth may be 
    lost, due to stealing, robbery and the like, and he has no way to protect it 
    other than putting it in a riba-based bank, for example, then he is granted 
    a concession allowing him to put it in the bank or other riba-based 
    institution, without any interest, so as to preserve it, because that is 
    committing the lesser of two evils. End quote. 

    Secondly:

     If putting the money in one account will pose a danger to 
    it, then there is no reason why you shouldn’t open another account – without 
    interest – after making sure of what the bank is claiming. If the danger can 
    only be removed by opening a savings account, that is permissible, so long 
    as one does not benefit from the interest and donates it to Muslim charities 
    instead, such as schools, hospitals, helping the poor and needy, and so on. 

    But you should not do this until you have made sure that this 
    danger is in fact real, as stated above, and made extra sure of that, 
    because of the severity of the prohibition on lending with riba, because the 
    Prophet  (peace and blessings of Allaah be upon him) cursed the one who 
    consumes riba and the one who pays it, and it is well known that savings 
    accounts are regarded as a loan from you to the bank, and this is a 
    riba-based loan. 

    And Allaah knows best.