The previous answers are correct in stating that you can indeed open a bank account prior to being issued a SSN. Further, there is not a legal restriction on non US citizens opening a bank account at a US-based bank. The primary regulations covering the opening of bank accounts in the US involve measures at countering terrorism and money laundering which place certain requirements with regard to the type of identification a person opening an account must present.
However, each institution may set specific requirements, above and beyond the legally required identification documents.
As stated above, you are not required to have a social security number to open a checking or savings account however many banks.require some form of identifying number. If this is the case these institutions will often accept a Taxpayer Identification Number (TIN) which a person without a social security number can obtain from the IRS.
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