Some answers here, including one by "charlie" suggest that the phrase "within its jurisdiction" which appears later in the 14th Amendment (in the Equal Protection clause) and which was at issue in Plyler v. Doe 457 U.S. 202 might have a significantly different meaning than the phrase "subject to the jurisdiction". which appears in the first sentence of the 14th Amendment, and which controls the citizenship issue.
In United States v. Wong Kim Ark, at p687, that Court very specifically addressed this point, saying:
The words "in the United States, and subject to the jurisdiction thereof" in the first sentence of the Fourteenth Amendment of the Constitution must be presumed to have been understood and intended by the Congress which proposed the Amendment, and by the legislatures which adopted it, in the same sense in which the like words had been used by Chief Justice Marshall in the well known case of The Exchange and as the equivalent of the words "within the limits and under the jurisdiction of the United States," and the converse of the words "out of the limits and jurisdiction of the United States" as habitually used in the naturalization acts. This presumption is confirmed by the use of the word "jurisdiction" in the last clause of the same section of the Fourteenth Amendment, which forbids any State to "deny to any person within its jurisdiction the equal protection of the laws." It is impossible to construe the words "subject to the jurisdiction thereof" in the opening sentence, as less comprehensive than the words "within its jurisdiction" in the concluding sentence of the same section; or to hold that persons "within the jurisdiction" of one of the States of the Union are not "subject to the jurisdiction of the United States."
Let me emphasize that last sentence:
It is impossible to construe the words "subject to the jurisdiction thereof" in the opening sentence, as less comprehensive than the words "within its jurisdiction" in the concluding sentence of the same section; or to hold that persons "within the jurisdiction" of one of the States of the Union are not "subject to the jurisdiction of the United States."
It seems to me that this makes it clear that the two phrases are simply different ways of saying the same thing, at least as far as the court in Wong Kim Ark held, and therefore the decision in Plyler v. Doe, that the children of those present in this country without lawful authority nonetheless had rights to education also is precedant that such children are citizens if born in the US.
The Plyer decision contains the statement that:
Appellants argue at the outset that undocumented aliens, because of their immigration status, are not "persons within the jurisdiction" of the State of Texas, and that they therefore have no right to the equal protection of Texas law. We reject this argument. Whatever his status under the immigration laws, an alien is surely a "person" in any ordinary sense of that term. Aliens, even aliens whose presence in this country is unlawful, have long been recognized as "persons" guaranteed due process of law by the Fifth and Fourteenth Amendments. Shaughnessv v. Mezei, 345 U.S. 206, 212 (1953); Wong Wing v. United States, 163 U.S. 228, 238 (1896); Yick Wo v. Hopkins, 118 U.S. 356, 369 (1886). Indeed, we have clearly held that the Fifth Amendment protects aliens whose presence in this country is unlawful from invidious discrimination by the Federal Government. Mathews v. Diaz, 426 U.S. 67, 77 (1976)
It goes on to say that:
In appellants' view, persons who have entered the United States illegally are not "within the jurisdiction" of a State even if they are present within a State's boundaries and subject to its laws. Neither our cases nor the logic of the Fourteenth Amendment support that constricting construction of the phrase "within its jurisdiction."
In view of the holding in Wong Kim Ark that
It is impossible to construe the words "subject to the jurisdiction thereof" in the opening sentence, as less comprehensive than the words "within its jurisdiction"
this seems pretty clearly to hold that such children are indeed "subject to the jurisdiction" of the United States, and so are citizens when born within the physical limits of the US.
Of course, all Supreme Court decisions are subject to change if the current court sees fit, but this seems a very large and unlikely change for a Court to make.