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ING is a Dutch-based global company (the N is for Netherlands) and while the rules may vary in each locality, I'd expect any global group like ING, HSBC, etc. to be a good start if you want to access funds globally. At least, a better start than some "local" bank with more limited resources.
In the US there are no requirements that you must be a citizen to open a bank account, but you will need to provide multiple IDs in any case. Not having a US Social Security number (banks are required to make reports and withholdings based on that) may complicate things, you might want to check with a bank.