Probably not possible unfortunately. The only thing that might be able to get you close is
Citibank IFS; if you're a serious heavy-hitter (I'm talking private-bank-customer grade) then they might be able to sort you out with a US-domiciled account.
Seriously, though - if you want a US bank account, just hop on a plane to San Francisco! The Chase branch at 101 Montgomery sorted me out in about an hour, and all they needed was a passport and two Singaporean utility bills (I used my SP Services bill and a Starhub bill) - you don't need a US address at all. A buddy of mine used some random Chase branch out in the suburbs and they were just as quick. Literally - just rock into any Chase branch in downtown San Francisco and you'll be in and out in under an hour with a US bank account.
If you end up doing this - or, for anyone else who stumbles across this thread - you want to go with Chase. Bank of America is awful; Citibank is meh; and Wells Fargo is good but Chase is better.
First Republic Bank is quite good as well (and they refund your ATM fees anywhere in the world, so you can use your First Republic ATM card in Singapore without running up any fees), but they don't have much presence outside the Bay Area.