I bet you cant due to the housing slump now they specially mention that you bring your green card. Even if you get an auto loan these day's they ask for your green card. I know two people one went to refinance he own that house since 3 years and when he tried to refinance they asked for his green card at the end of the process. The second guy went to buy a car and he too was asked for a green card. I think it also depends on where you live but 80% of the lenders are trying to make sure you have some ties with US. So that you dont perish after you get the approval which happened in most of the case, So  my suggestion is wait till it gets approved. or confront the lender about your status and ask them in written that they would not make judgment based on your legal status before you apply. They brokers or the real state agents would say yes yes that's fine but in the end it wont go through.

Best Wishes,
YekloYatri