Depending on regulation and international politics, this can eventually take place in the future. When global authority identify Bitcoin as a legal means of exchange, then banks will have to follow if they must stay relevant. Presently some banks view it more as just asset than money. Adoption will only grow if government provide obvious rules and users continue to request it during transactions.
Banks do not just view Bitcoin as an asset, but they consider Bitcoin and its children as a revival. They believe Bitcoin is taking food off their table. If they have their way to make things go sideways for Bitcoin, they will even do it to make sure there is nothing like cryptocurrency. For international law to start pushing for Bitcoin global regulations, the majority of the countries will also have to support it before such a move can even be thought of.