Instead of going on credit score and income, they will go by your relationship with them and the amount of money you have on deposit with them. Several of my friends have done this while living/working in the US without becoming a citizen or obtaining a SS#. What you cannot do is use any online application since all will ask for SS#. My advice would be to go to the bank where you hold accounts and talk with one of the banking associates.
The problems usually only start if you get an account that earns interest, because then the immigration status has to be completely defined.
But yes, US laws in this field are strange. I came to the US on a J-1 visa, which is a visa that doesn't allow you to work. Nevertheless, you get some compensation for your "work". In order for your host (a state institution in my case) to pay this compensation (well, the only difference to a salary is what it's called), you need a SSN. Which means that the Social Security Administration has to give you an SSN with a visa that doesn't allow you to work. They do this, but not all frontline employees know this (took me some time to convince them to take my application, by pointing out that they really don't have to decide anything by themselves).
On another note, my main bank still has problems with my account, because they don't know the difference between an ITIN and an SSN (as all employees say the same, I guess that's the result of their faulty training). It doesn't help that the term "US resident" has several definitions. For the IRS, I am an US resident. For the DHS, I'm not. Unfortunately, when filling out forms, nobody defines what definition of "US resident" they mean.
Just to make sure: I'm here completely legally, with a work visa of the non-immigrant type.
I am tickled by that icon of yours...mind if I ask where you got it? oops! sorry, I forgot this site is about credit.
I qualified for a secured card and for the savings account they sent the form for non us citizens tax id number by mistake. I had to go get the us citizen equivalent from the IRS site. They must have the non citizen forms for a reason.
The preqaulification is pretty easy. Then they send a large packet of forms.