Really? We're still on this?
Once upon a time, my brother applied for a government job working in a warehouse and underwent a rigorous background check from the U.S. government that lasted a bit less than a year. They found everything out about him from his birthday, to the day my dad purchased our first car, to his social networks, to the specifics of his work history.
That being said, I think it's a safe bet that the same government can find out whether or not The President of the United States is an American.