Only naturalized citizens, and then only by granting congress the power of naturalization.
Natural born citizens were created by the Declaration of Independence, the document which created the country.
Seriously, you can't have citizens until you have an independent country. That Independence document is the one that created the country, and therefore "citizens."
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
That establishes citizenship at birth. Congress can grant extra situations in which someone is granted citizenship at birth, but the Constitution does indeed have a say in this issue.
Congress is given full authority to determine naturalization procedure and law, but the only two forms of citizenship that is contemplated by the Constitution, according to the Supreme Court, are citizenship by birth (whether given by the Constitution or by laws of Congress) and citizenship by naturalization.
There is not a third form of citizenship.