First of all I dont think so that this will ever happen. Bitcoin being decentralized aims at eliminating intermediaries involved in financial transactions. This makes banks totally useless. Why would banks be so welcoming and friendly towards Bitcoin under these situations? Second, even if it accepts such deposits, it would not be very attractive. I mean I will not deposit my Bitcoins there as I can earn much from the market than a small, stable rate provided by banks.
That's one interesting point you've presented here, my friend. There would be no reason for banks to offer Bitcoin deposits as it would defeat their real purpose in this world. Since Bitcoin eliminates the middleman, it's often seen more as a threat to banks and governments alike. The level of decentralization the original cryptocurrency has, allows anyone to be their own bank.
Another thing that would need to be considered is that Bitcoin is an extremely volatile currency, when compared against fiat currency. Its instability would render deposits in banks useless, as they won't be able to offer stable interest rates to their customers. Therefore, I think that it's best for people to secure their own Bitcoin by being their own bank. There are many alternative cryptocurrencies which offer attractive stake rates, allowing you to multiply your money without the need for a bank. As such, Bitcoin is the cryptocurrency you'll ever need to gain control of your financial future.
