Banks like Bank of America, Wells-Fargo, JP Morgan Chase, etc. all have a merchant services application process that you can apply for. There is quite a bit of documentation that needs to be provided for to qualify as a master merchant, i.e. a PSP.
The main issue here would be to demonstrate your business model and how you will mitigate risk for the bank. This is not overly difficult if you're in the know-how/business, however, you might also want to consider that, as an option, you can also go to large-scale processors like First-Data, MIGS, Cybersource, Chase Payment Tech, et. al. to obtain the same. Many a times, these processors offer added layers, that a bank might not be able to offer (not always the case, but nonetheless an important differentiator).
One of the most important aspects on this is to understand that paperwork aside, you will need to negotiate your way into getting a better MDR, reduce your risk deposits and get better payout terms for yourself.