I'm a traditional lender who sell all or our loans (needing to meet Fannie/Freddie guidelines), and have confirmed these agencies do require social security #'s....
It's possible that the few lenders that retain their loan (portfolio their loans) would be able to lend to someone with just a ITIN, but you may run into other complications like getting insurance coverage which is required in this process...
Let us know if you've been successful!
Best wishes, Jeff M
If you have any further questions or if you would like a loan, feel free to contact me through the information on my profile page!
Hope this helps and good luck!
Sr. Mortgage Banker
NMLS # 659743
RE/MAX Active Realty.
Phone: (510) 676-4770
Sarah Y Jaime
Keller Williams Realty