You can open bank accounts from outside the US, with no need to visit in person. And, furthermore, you do not need to have a US address or SSN.

Non US resident bank accounts offer advantages for non-us citizens and residents. These are principally those who conduct business globally or online, and wish to send/receive payments in US dollars.

Furthermore, for philosophical, as well as privacy concerns we would recommend that clients minimize their exposure to US banking whenever possible.

It's not possible for you to open a normal US bank account online, if you are a US non-resident. Often you'll have to contact the bank from their online form and they'll then contact you.

In almost all cases you'll be asked for a letter of reference from your local bank and you'll have to submit a copy of a bank statement, rent receipt, utility bill, or government issued identification which shows your address in your country of residence and a copy of your US accepted passport.

I can't say these US banks listed here are "asking" for US non-residents clients, as it takes time to browse each site to find if they offer International banking or US non-resident accounts. However, you will be able to visit their International bank pages in the listings as I have done all the homework for you.