Yes they will accepted it as a payment method and system,just like here in my country which is the Philippines our Philippines National Banks was totally looking for a bitcoin and currently many banks here was looking to use a blockchain that can we use for fast transaction.
My answer is yes. Some already accept btc, but some are still in the process of studying the advantages and disadvantages on adding bitcoin on their banking system. For me i agree on this thing, it will be more easier for us bitcoin users to deposits and to withdraw, i think it will be more secure if you store your bitcoin in banks than in online wallets.