Acts of the Parliament of England/Great Britain define who may be a natural born subject, not the "common law" of that country. The definition has varied over time.
Acts of the Congress of the United States defines who may be naturalized and by what mechanism. Naturalization creates citizens, Congress has no power over creating natural born citizens.
Congress cannot make someone a NBC, but they probably can define its meaning. That is based on the English common law principle which had ‘natural born subjects’ who both met the original definition, and also those who had been defined as such by Parliament.
“Of relevance to any meaning and “common understanding” of the term “natural born” within the American colonies and at the time of the drafting of the Constitution is the legal treatise on the laws of England referred to as “Blackstone,” for its author William Blackstone. Published in 1765, this treatise was not only available, but was widely known to the framers at the time of the drafting of the Constitution.
As noted by the Supreme Court of the United States, “Blackstone’s Commentaries was widely circulated in the Colonies ...,” and that “undoubtedly the framers of the Constitution were familiar with it.” As discussed in the earlier section of this report on the common law, Blackstone explained that “natural born” subjects in England and the American colonies included all those born “in” the lands under British sovereignty. Concerning specifically the issue of children born abroadof English subjects, Blackstone explains clearly that such children are then (in 1765) considered under the law of England as “natural born” subjects, and have been considered as such for most purposes since at least the time of Edward III (1350), because of the development of statutory law in England to “encourage also foreign commerce.”
As stated by Blackstone in his 1765 treatise,
[A]ll children, born out of the king’s ligeance, whose fathers were natural-born subjects, are now natural born subjects themselves, to all intents and purposes, without any exception; unless their said fathers were attainted, or banished beyond sea, for high treason; or were then in the service of a prince at enmity with Great Britain.
The “commonly understood” meaning of the term “natural born” in the United States at the time of the drafting of the Constitution might thus be broader than the early, strict English “common law” meaning of that term.
As noted by Charles Gordon, former Chief Counsel of the Immigration and Naturalization Service, whether the body of English law in the 1770s was from early common law, from statutory law, or from the common law modified over the years by statutory law, these provisions “were part of the corpus of the English law in existence at the time of the Revolution, which was substantially recognized and adopted by our forefathers.”
This common usage and popular understanding to the framers of the term “natural born” subject (as employed in England), and the term’s apparent evolution and broadening of meaning through statutory law, has thus led several other legal commentators and historians to conclude: “The constitutional Framers had a broad view of the term ‘natural-born’ and considered all foreign-born children of American citizen parents eligible for the Office of the Presidency”; or, as stated by another: “[T]he delegates meant to apply the evolved, broader common law meaning of the term when they included it in the presidential qualifications clause.”
Presidential historian Michael Nelson has also averred that the term appeared to have a common meaning at the time of the drafting of the Constitution which involved within its concept both the common lawdefinition and mode of acquisition of citizenship (through jus soli), as well as the common understanding of the long-standing broadening of such term by the operation of English statutory law to include those subjects who may have traveled abroad for purposes of commerce, or otherwise. As noted by Nelson (and pointed out by others), a more restrictive meaning to include only those born within the boundaries of the United States would mean that John Jay, who may have recommended the precise term to the Convention, would have intended to exclude from eligibility his own children who were born in Spain and France while Jay was representing the United States abroad:
The provision for “natural born Citizen” probably was aimed at immigrants, although the term is so unusual as to be vague.... [b]ut [it] had deep roots in British common law. In medieval times it had embodied the doctrine of jus soli: a natural born citizen was one born within the realm (on the soil, so to speak). But with increased commerce and travel, Parliament, starting in 1350, seemed to expand the definition of natural born to incorporate the doctrine of jus sanguinis. Now babies born of British citizens abroad or at sea were included as well. One can presume only that Jay and the delegates meant to apply the evolved, broader common law meaning of the term when they included it in the presidential qualifications clause. Certainly Jay did not mean to bar his own children born in Spain and France while he was on diplomatic assignments, from legal eligibility to the presidency.”
http://www.fas.org/sgp/crs/misc/R42097.pdf