I'm not sure I understand what you're asking. Do you want to set up direct deposit to a US-based Chase account from the US, or from the UK? And you'd like to pay rent in the US, or in the UK, by automatic deductions from that US-based account?
If it helps, this is my set-up: I have a US bank account where I receive a US-based pension by direct deposit. I also have Netflix billed to that account and I make a monthly donation to my old local NPR station (even though I will probably never listen to it again...
). I also have a UK current account and savings account. I move money from my US account to my UK account using Transferwise as needed.
I use the debit card from my UK account for POS transactions here because it has chip & PIN, and US ones still don't (
). I don't have any UK credit established so if I need to make a credit purchase, I use a US-based credit card. My credit card banks have stopped charging minimum foreign transaction fees and they use the XE rate + 1% (so slightly more expensive than Transferwise, but nothing like it used to be).
I feel like I haven't answered your situation, though.