According to Romney family lawyers it doesn't matter if Barack Obama was born in Hawaii, Kenya, or even Paris: Because his mother was an American (and not even Donald Trump questions that), he is eligible to be President.
The Romney lawyers investigated this question in the 1960s, when Mitt Romney's father, Governor George Romney of Michigan, was vying for the Republican presidential nomination. George Romney had been born in a Mormon colony in Chihuahua, Mexico, as his grandfather moved there with his wives in the 1880s after polygamy was outlawed in the U.S.
While some opponents nicknamed him "Chihuahua George," his suitability for the highest office because of his birth was never seriously challenged. The reason his campaign faltered was because of his shift in position on the Vietnam War: He went from being a supporter to opposing it, infamously claiming to have been "brainwashed" by military officials. After that Richard Nixon's lead in the polls more than doubled.
Article II of the U.S. Constitution states that "No person except a natural born citizen … shall be eligible to the Office of President," and so the question is: Does natural born citizen mean born a citizen or born in the U.S.? The Founding Fathers were of course aware of both jus soli (birthright citizenship) and jus sanguinis (citizenship through parentage), but deliberately wrote "natural born" rather than something like "born on U.S. soil," arguably to include children born to U.S parents outside the country.