Doing a background check without a Social Security number from a rental can often be done. It's because of the interest paid on security deposits.The good news is that you do not need an SSN on an apartment rental application to screen a prospective tenant. I seen the apartment twice so Is it safe to fill it out ? Img. logo. Lawyer: GAL. If the applicant does not have a social security number, you generally can't run a credit report or nationwide criminal check. The Rental Application. The property manager or landlord may ask a prospective renter to fill out a rental application. You should be able to rent an apartment without a Social Security number provided you have a student visa or Permanent Resident Card (a.k.a. A landlord may ask you to fill out a rental application.