Depending on the market, a bank is going to look at a combination of low-cost customer acquisition (particularly in strategic growth segments), mass-market desposit mobilization, credit portfolio growth and fee-based income. The respective weight of these, in terms of relative importance, will depend on the bank and its strategic objectives. It will also vary based in whether you are talking to an acquiring or an issuing bank. The best way to approach a bank is therefore to identify which core business element your startup is best positioned to support, and which bank is likelier to find the value prop attractive. I would suggest talking to the head of retail banking, the head of credit business and the head of acquiring business.

To answer those questions best you should frame those questions around the specific problems you are solving.
What problems are we solving for the bank?
What problems are we solving for the consumers of the bank?
How are the bank currently solving those problems?
In what ways are we significantly helping the bank and it's consumers?
Based on how you monetize and how you add value to everybody involved gives you insight into how you put together a partnership that not only generates interest but also has the potential to last.

Remember this: Triple Win Partnerships. A win for the bank. A win for your company. A win for the end user/customer. If all three win now and in the long term it could be a good deal. If it's short on any of the three, the deal won't last.

Start calling banks that you can quickly and easily call and meet in person if there is an interest in your product and service. Start NOW if you haven't already. The best direction you will get is from the people you speak from the bank yourself. You will also learn a lot going through the process. Time to suck it up and make some calls.

Last note: the example you gave... Stripe and Wells Fargo... Based on the other partnerships I've established with banks/payment processors, they are always open to partnerships that bring in new merchant accounts that are low risk and profitable. In this case specifically, it's more than likely a shared percentage of the net revenue on fees collected from the merchants. The shared amount between Stripe and Wells is probably 95% Stripe to 5% Wells. Also, Wells probably doesn't do a whole lot. They more than likely are just the sponsoring bank for ISO registration to Visa/MasterCard.

I hope that helps. Let me know if there is anything else you need.

Here to serve,

Patrick