The first sentence of the 14th Amendment to the Constitution -- often called the "citizenship clause" -- reads as follows:
"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."
The original intent of this provision was to guarantee citizenship to the former slaves and their descendants following the Civil War. However, the Supreme Court held in U.S. v. Wong Kim Ark (1898) that the "citizenship clause" applied to anyone born in the US, of any ethnicity or national origin -- and, in Afroyim v. Rusk (1967) and Vance v. Terrazas (1980), that the clause prevented Congress from revoking a person's US citizenship without evidence of his or her intent to give up said citizenship.
That only dealt with legal immigrants. There has never been a case to my knowledge that was specifically ruled on as it related to illegals. In any event since Congress has legislative authority over immigration matters they could write a law around the jurisdiction clause denying automatic citizenship then let the USSC hear it when it's challenged.