As of now, there’s no Bank that would let you do so. Banks are not wallets that can hold your Bitcoins.
However, there are platforms like Payza that would allow you to exchange your Bitcoins for fiat and withdraw it as your local currency to your Bank a/c, Credit Card etc.
Payza also allows you to buy Bitcoin and store it, in addition Payza accepts number of altcoins as deposits. The process it pretty much simple.
Open an account with Payza, transfer your Bitcoins to Payza, link your Bank and withdraw as your desired currency to your Bank a/c.