Having a US bank account is not a condition for having a US-issued credit card. In Europe many credit cards directly debit your bank account each month for the balance due, thus you need a bank account. This is not true i the US. You may choose to have the credit card company debit your bank account to pay the credit card but it is not a requirement.

However, to obtain a US-issued credit card you do need a SSN, which you have, and usually a US address, which you might or might not have. You also need a credit history through one or more of the 3 credit reporting agencies, which you might or might not have.

If I were you and I were interested in obtaining a US-issued credit card, I would try applying for one and see what happens.

Good luck.