OK, Bill. From experience, all one does is apply for a loan. The banks do not ask about your immigration status, all they need is a SSN! The banks are NOT the Department of Homeland Security. It really is that simple
When I was a Mortgage Banker and when an undocumented alien came to my office to get a loan, they were not required to give us a Social Security Number. However, if you were a Legal Citizen, you were required. Go Figure!
Lenders can approve or disapprove of any mortgage application they want regardless of credit or citizenship status. And yes, it is that simple.