If there is any bank that allows deposits in Bitcoin, I am not aware of it. But I believe it will not be something very easy to find. This would go against the interests of the banks, in addition they would be promoting a technology that would make the users decrease the dependence on banks, since Bitcoin wallets allow the users to have a greater control over their money.
nothing will decrease the dependence on banks, at least not in my opinion...
even if someday bitcoin achieve massive adoption, the majority of users will be of people who dont know nothing about technology.
these people will avoid to have a personal wallet/node, because they dont know how it works and that's where the banks will help. bitcoin is a currency and banks manage currencies so to me it's just a matter of time to banks start to use bitcoin.