First of all, you need a bank, which is willing to give you an account. To provide banking services to non US citizens, non residents is not a right you may ask from a bank, but is a privilege given to you from that bank. So whatever the bank asks from you counts. Discussing about this with the bank may not be useful.
You may look for a larger bank, e.g.
- Bank of America
- JP Morgan Chase
- Wells Fargo
- Citibank
Some smaller and specialized banks and credit unions may also provide banking services to you, but it may be harder to get.
What is the stepping stone to get a bank account?
As as result of sloppy bank procedures before the 9/11 terror attacks, which enabled terrorist to use banking services as part of their terrorist activities, requirements have been tightened mainly by the so called PATRIOT ACT.
Besides this most banks fear legal ramifications from law enforcement and regulatory authorities, and may have their own understanding of these requirements. So banks may have even higher requirements than what is actually required by law.