There is at a U.S. bank (we work with them a lot) that will open business account for a non-US business entity.
Citibank, Chase, Wells Fargo will only will only open an account for a a U.S. entity but it could be owned or controlled by a non-resident.
Qualifying/registering a foreign company in the U.S. won't do it for these banks.
Generally, registering a foreign company in the U.S. (in some states it is referred to as receiving Authority to Transact, in others receive Foreign Qualification, etc.) is tricky from the taxation point of view. Such US presnce may well be regarded by the IRS as a branch of a foreign corporation.
Lastly, if you are looking for a personal account a number of banks will open one for you as a non-resident.