State laws can vary quite a bit, so thanks for mentioning you're looking in VA.
Looking at the Virginia Landlord-Tenant handbook (link below), and reading the entry under the rental application, it does not specify that the landlord can ask that question...so you're not required to provide it as far as the state is concerned.
On the other hand, it does not specify he can't ask for it either. If you don't provide it, he may not accept you.
So what to do?
When I found myself in your situation (here in Calif.), I simply provided the checking account number and bank name. (which he'd get from the checks I paid rent with anyway). No balances, no statements, no savings info. It was enough for my prospective landlords, and not so much that I was uncomfortable.
If he's not collecting an application fee, fill in only what you feel comfortable with and submit it. If he is collecting a fee, talk with him about it before you submit the app. If he's going to automatically reject it without that info, and you don't want to provide it, go somewhere else.
My strongest advice is to consider how you get along with your landlord BEFORE you sign a lease or move in. If he's not reasonable when you're applying, he's probably not going to suddenly become reasonable after you move in. Think of the application process as an interview - is this someone you want to be in business with?