In Article I, Section 8, that specifies the powers of Congress. Clause 4 gives Congress the power "To establish an uniform Rule of Naturalization..." So Congress decides who can and who can't become a citizen. And it's backed up by the role of gubment that's spelled out in the preamble of the constitution. "Form a more perfect union, establish Justice, insure domestic Tranquillity, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity." It can be argued that who can or can't become a citizen is pretty tied to almost all of those.