TimFountain

I got a mortgage in 2001 on an L1A with a letter from the company immigration lawyers stating that they were starting the GC process. i.e. no sign of an actual green card at that point. Oh and this was from WF, so it it for sure possible. On difference might be the downpaymnet. We put down 80% and had more than enough in documented savings to pay off the balance at any time. We only did it to establish credit. We paid it off over 3 years.

Pulaski

I don't think it's the visa v green card that is the stumbling block as there have been a number of examples over the years of people on work (non-immigrant) visas getting a mortgage.

The issue is likely either the green card as evidence of someone's immigrant status (as per the OP in this thread), or, that some visas, notably some E-2's, are held by people who are effectively self-employed, and that puts them in a different situation with respect to their creditworthiness and eligibility for a mortgage.