I have responded to this question before (from someone else) however I'll be sure to be more specific this time.
No lender that I'm aware of will do loans with only ITIN numbers anymore. TCF Bank, 2nd Federal Savings, and Banco Popular used to do them, but they have all discontinued writing these loans.
We have a program for foreign nationals. An ITIN and SS # are not required. Neither is a US credit score.
Here are the requirements:
1) Must be employed (or have an offer letter)
2) Must have an authorized work visa
3) Minimum down payment of 20%
This program is available in all 50 states. I can do loans in Illinois, Wisconsin, Indiana, and Michigan. If you're in another state, I can refer you to the broker who can do loans in your state.
If you have any questions, please let me know.
Senior Loan Officer
Chicago Financial Services
312-204-6556 - direct
312-608-1555 - mobile