Unlike employers, landlords are not restricted from renting to undocumented residents of the United States or those without social security numbers. In fact there are many people in Texas that fall under this category. In general there are two types of residents without social security numbers: documented and undocumented.
Examples of documented residents that may not have a social security number include those with green cards, work visas, work permits, or I90 student visas. For these types of renters there is a supplemental non-resident application that you will need to fill out. With this type of application, apartments will also require legal documents verifying your status within the United States. These residents often do not have verifiable income from an employer within the United States. If that is the case it is important that you are able to show money in a US bank account, whether the money is wired in from your country of origin or from a student loan.
Undocumented residents are those that have no social security number and no documentation of their status within the United States. Since criminal and rental history checks are much more complicated for these types of residents, very few apartments will rent to them. Those that will usually require an additional deposit that can be as much as 1 month’s rent. All will require some form of ID, like a Matricula (if your country of origin is Mexico), or a driver’s license or resident card from another country. They will also require you to verify your income.