While looking for something on the web I came across this document from May 2006: Financial Access for Immigrants: Lessons from Diverse Perspectives published by Federal Reserve Bank of Chicago.
Pages 46-50 are especially interesting as they concern mortgages issued with ITIN only. Fortunately, Fannie Mae would not touch ITIN-only mortgages, but other companies did purchase them. For example Hispanic National Mortgage Association would buy them and resell as securities.
Here is a sample quote (highlights mine):
Although the movement is in its infancy, a small number of banks have pioneered home mortgages using ITINs for people without Social Security numbers. Small banks have been the innovators in this arena, but large banks and sub-prime lenders have begun to issue ITIN mortgages as well (see Textbox 10). Some banks are motivated by the strong demand for home loans from undocumented immigrants. Others view ITIN mortgages as a mechanism for expanding their businesses if demand for traditional mortgages begins to slow.
… Many bankers and community-based organizations view the development of ITIN mortgages as consistent with good business practice and with good community development policy. These loans offer banks an opportunity to increase the size of their lending portfolios. Critics of ITIN mortgages condemn the practice of making it easier for immigrants who have violated federal immigration law to own homes in the United States. The fact that most of these loans cannot be sold on the secondary market has also made many lenders skeptical about issuing them. Both sides of the debate are looking to lawmakers and regulators to provide greater clarity on this issue.
Overall, this document seems to be positive about the ITIN-only mortgages, maybe hoping that future regulation would expand that market. Freddie Mac was hinting at wanting to start buying ITIN-only mortgages, see Plan would ease ownership limits for illegal immigrants. Here is a quote:
Indeed, a Freddie Mac spokesman says his company may soon be willing to buy these loans as part pf a broader test.
If the big mortgage insurer’s pilot program proves successful, and Freddie Mac determines that these loans are no more risky than the other mortgages it purchases, the number of lenders serving ITIN borrowers will increase and the cost of ownership for ITIN borrowers will come down.
It looks like banks were willing to sell any type mortgage as long Fannie Mae or Freddie Mac were buying them in the secondary market.