Residence status has no bearing on whether you are eligible for a mortgage, or if you are eligible for a good interest rate. Only your credit & employment history, and income are important.
The reason they ask about residence is because in the old days they could quickly filter out someone who has not been here long enough to establish a strong credit history. In most cases these days, this assumption by the lender is incorrect, as people have spent years living here on H1Bs, and have very good credit, but the lenders are still basing thier knowledge on old facts of immigration.
Therefor I suggest that you ask what the credit score is that they require for the mortgage, and be sure to know your credit score when you are shopping around so that they can't just casually deny you on the basis of not having a green card.