I know a bit off topic, but I'm confirming as well - my bank, Trustmark (a southern regional bank) accepts foreign-held accounts without residency, but you have to go in person. You can't do it over the internet.
Also, I know for sure that Wells Fargo will accept foreign accounts as well. My ex-partner had a good relationship with his particular branch and they actually set up bank accounts for our programmers in the States without them having to go to the States, but sponsored by my ex-partner.
I suspect there are few banks in the States that will accept just anyone that does not have residency in the states opening up an account electronically, without going in person, at this point.