Due to the situation of a friend, I asked at my local bank in the US about mortgages for immigrants. (This was, admittedly, before all heck broke loose last fall - but all my bank statements and my local bank website have been promoting the fact that "we are still making mortgage loans!)

My local bank is a small, independent bank. They will make mortgage loans to non-citizens who have visas. Standard mortgages are available to green card holders, but for those with other categories of visas, they will consider making loans for the duration of their visa terms. Obviously, they are only going to make a loan on property located within their area.

I would be very wary of mortgage brokers (especially in the US) as they get paid to get the loan accepted and we've seen the unfortunate consequences of that practice. Try working with a local bank - not one of the big chains - and see what you can arrange. Obviously, they are interested in getting your overall banking business - checking and savings accounts, credit cards, etc. - and have every reason to work with you rather than just for a one-time commission.
Cheers,
Bev