The singular focus landlords have with tenant sceening overlooks 2 very important things.
1. The intended purpose of an SSN - i.e the reason the government gives you one. It's not to support tenant screening.
2. By the time we assemble an SSN and all the other things we demand from a tenant we have put ourselves in position to steal their identities.
So here is a list of things you can do without knowing an SSN.
1. Look up a persons background in the court system.
2. Check their financial standing by looking at their bank statements.
3. Talk to employers and previous landlords.
4. Visit them in their home to see how they currently live.
5. Look at their paystubs.
In fact I'd argue that you don't even need an SSN to run a credit check. Personally if I've done the aforementioned I don't think running a credit check is going to tell me anything more, so here's what I do.
1. I tell them the application fee is for running the credit check because we have to pay for it and I tell them that if we reject the application before running the credit check we will refund the application fee.
2. I tell them I disqualify applicants who give me dishonest answers.
3. I ask them what I would see if I ran their credit.
For the information of others on the thread - you can have an SSN without being a permanent resident much less a citizen. I got one by being an international student on study abroad in the US.