Residential Lending in U.S. to citizens of other countries:
For conventional financing, there are documentation requirements for your legal status. If you have a Social Security Number and a Green Card, you have satisfied that requirement (Permanent Resident). For non-permanent residents, you still need an SSN, and either a VISA evidencing legal entry to United States for temporary residence, or a passport and an Employment Authorization Document.
You must have documentation of legal entry into the U.S.
Special rules apply if your income is passive only (not self-employment or salaried).
The documentation guidelines above are universal. Different lenders will add more restrictions than are shown here.