It is an established maxim that birth is a criterion of allegiance. Birth however derives its force sometimes from place and sometimes from parentage, but in general place is the most certain criterion; it is what applies in the United States; it will therefore be unnecessary to investigate any other.
James Madison, House of Representatives, May 22, 1789
I don’t have a lot of time to dig further into this right now but if that quotation is all that matters then any child born to a mother who managed to illegally sneak into the country long enough to give birth is qualified for the presidency. Regardless of today’s goings on I have a hard time believing that is what the founders intended. Certainly they didn’t consider everyone born in this country natural born citizens, that would have meant that slaves as well as all the members of the native american tribes would have been eligible to the presidency which is obviously not the case.