I believe the constitution forbids dual citizenship for a President just for that reason.
Where does the Constitution say that?
The Constitution does not address the situation.
BTW, most folks with dual citizenship, or who could claim dual citizenship, are unaware of the fact, or they've never asserted their right, and probably don't care.
George Washington was, according to British law, a UK subject. He was also a citizen of Virginia per the Treaty of Paris. Later on, he became a citizen of the United States.
If you carefully read the Constitution you will find that ALL AMERICANS are citizens of a state and of the nation as a whole. We are born with dual citizenship.