[Panos]

This is a regulatory requirement, part of the "Know Your Customer" doctrine. In plain words, banks are required to know who is the client who has opened an account.

Most banks will be risk averse and will not open an account to anyone applying online from abroad. Even for US persons applying online, they will ask quite a bit of documentation.

Some banks (but not all) will open an account for a non-US person, when the non-US person physically visits a US branch, with proper identification (typically a passport) and documentation on why they want the account. But even in such cases, it is up to the discretion of the bank employee to decide whether the risk of opening an account for a non-US person is worth the benefit. So, the same bank may give different replies to the same inquiry, depending on the branch asked.

As a concrete example, TD Ameritrade will open easily an account for a foreigner in the Chinatown branch in NYC, but will not open an account when the same customer visits a branch in midtown in NYC.