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You can open the account without an SSN. However most banks will generally require you to provide a valid SSN or TN with a certain period of time otherwise they will close the account.

Generally if you are in the US legally you will have one of those numbers.

I got my bank account before getting my SSN. They ask for passport and maybe some other documents (e.g. I-20 for F-1 Visas) but certainly getting a bank account without SSN is possible.

I got my bank account with just my passport.