The application sounds legitimate. If the landlord obtains your credit report, the landlord must give you a copy of the report if you request it. If you don't feel comfortable, you may want to look for another rental unit.
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, consult with your own attorney.
First, use by a landlord of social security numbers or similar numbers in the context of a rental agreement will be covered more by state, rather than federal laws, so I would recommend checking with a California attorney. Second, in other jurisdictions, its fairly common for a landlord to request information to verify your credit rating, as he or she makes the decision to rent to you or not. However, two things to remember: One, although I have heard of requests for social security numbers to verify credit, I have not heard of requests for bank account numbers, as normally this information is not needed (of course, the landlord often provides a bank account number for a renter to use to make deposits). Two, if the landlord does request this information, he or she must still protect it, and cannot generally use it for other purposes or provide it to other parties without disclosing this to you and following certain rules or procedures. The rules, protections, and procedures all vary by state, but most states (and without providing legal advice, I am fairly confident California would have some of the most protective laws regarding use of this information) have them and provide various private rights of action for violations. Whether you consult with a local attorney or not is up to you, but if the landlord is requesting such information without providing any disclosures on how he or she planned on using or protecting that information, I would definitely be concerned and contact a local consumer protection attorney.