I am looking at the same. I do not have an SSN/ITIN. I have to open accounts with my passport and ID documents. I am waiting to see if I will get my Chase bonus & my Bank of America bonus.
My brother is a US Permanent Resident with an SSN, and we had signed up for the American Express Checking for the $250; awaiting the qualification email, hopefully.
He already had the C1, so he cannot open another one and claim this bonus.
There are a few options out there. We got a Chase pamphlet today where you can get $225 with a bonus code, and the normal is $200.
There are a few options out there, but they do require big deposits. Another could be Ally Bank.
I am actively looking for those with a $0 fee.