First,
HSBC claims
The HSBC International Banking Center provides customers options to open personal deposit accounts outside the U.S. as well as inside the U.S. for non-U.S. residents.
Non-Premier customers requesting to open an international banking account outside the U.S. and requiring assistance in the completion of all necessary paperwork from the International Banking Center in the U.S. or an HSBC branch in the U.S. will be charged a one-time fee of $200 USD.
This might be the only option that can be done without visiting the USA. If you are visiting, your options are wide open.
The newest blog post attesting to being able to open a bank account as a tourist I am aware of is from 2017 February, How To Open A Bank Account in U.S., As A Non-resident Non - citizen, Wells Fargo Bank where a citizen of Georgia just randomly walks into a few bank branches in Manhattan:
it's not a problem to open a bank account at Citibank without permanent address in the U.S. but they would need some additional proof of address from my country, like some bill with my name on it and the address.
Wells Fargo bank, I decided to enter and just ask my options. And then the surprise come - sure, buddy we can open a bank account for you, no matter which country's citizen you are, and we don't need a proof of your home country's address, as long as you can provide us with a debt card info issued at your home country.
Luckily I had a Visa card issued by Bank of Georgia, and that's it - the process took less than 15 minutes, I had an active bank account with my name and my address in Georgia, the bank's clerk issued an temporary bank card, and told that after some 5-7 days I will receive my permanent card in Georgia by mail.
Wells Fargo didn't asked for ITIN when opening a bank account, but I'm aware that I should submit it at some stage.
Opening a bank account at Wells Fargo is free, there is no monthly fee, if on your account is $1,500 or more, otherwise it's some $10 per month
This page is a veritable treasure trove of relevant information.
The Bank on San Francisco program from 2016 September mentions that Bank of America will accept any foreign passport as an ID. I would call whatever branch you plan to visit but it's not unlikely you will succeed. I just tried the online application with a fictional citizenship-residency combo with the state set to Delaware and it invited me to visit them personally stating that the online applications require an SSN -- and not that all apps require an SSN:
