Hi Cheryl -
There are lenders who will do mortgages with an ITIN, though you're more likely to find them through brokers and credit unions than regular banks. Down payment requirements of 10-20 percent are common and expect to pay a higher interest rate than you would on a typical mortgage. Most lenders will also require a valid visa or work permit and two years of tax returns or income documentation. Credit score requirements vary; some lenders will allow alternative credit scoring that takes into account business ownership or other alternative credit factors.
We have a large number of lenders in our network and requirements among them vary. You can use the form on our home page to request rate quotes from them without a SS#.