There is no law that allows lenders to deny a loan, or to raise the interest rate, or demand more down payment money, just because of a person's immigration status. Actually there is a federal law called the "equality act", which prohibits this.
Any lender demanding more down payment, or threatening a higher interest rate, or simply denying the loan all together, is breaking the law, if they are making this decision based on your immigrant status.
If you find a lender who does this, then go to another lender, or tell the lender that you will report him to the better business bureau. They have no right to demand 20% down, or something similar, just because you do not have a green card. Many uninformed immigrants will fall for this trick.
The lender must decide on you eligibility based on your work history, income, and credit rating only.
In many cases people who do not have permanent resident status also have a short credit history and therefore a low score). Also these same people often have a short employment history. Ans as with Us citizens, people who fit this criteria are asked to give a larger downpayment, pay a larger interest rate, and in somie cases, are considered too high a risk to give the loan to. This is legal, since it is based on the person's financial status, not thier immigrant status.
But as immigration takes such a bloody long time in this country, many people have established considerable work and credit histories in the interim. These people would be eligible for the same best rates, and lowest down payments that any permanent resident or US citizen with a good credit history, and good work history could get.
After realizing this rule, my sister recently got a mortgage at 5.85% with 0% for 30 years. She has been here for 4 years, and she is in adjustee status.
I am not a lawyer, this is only my personal experience, and research.