I don't believe that it would require a Constitutional amendment to deny the children of illegal aliens citizenship. The notes debates from the ratification make it pretty clear that the amendment was not supposed to cover the children of those who are subject to the citizenship of another nation. Native Americans were born in the US but weren't granted automatic citizenship until the 1920s. The children of diplomats born in the United States are still not granted citizenship, even though they are here legally.
the Fourteenth Amendment states, "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."
There is an article about this topic on numbersusa.com that says, "The jurisdiction requirement was added to the original draft of the Fourteenth Amendment by the Senate after a lengthy and acrimonious debate. In fact, Senator Jacob Merritt Howard of Michigan proposed the addition of the phrase specifically because he wanted to make clear that the simple accident of birth in the United States was not sufficient to justify citizenship. Sen. Howard noted that the jurisdiction requirement is "simply declaratory of what I regard as the law of the land already." Sen. Howard said that "this will not, of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the Government of the United States, but will include every other class of persons."
If you are interested in the reset of the article you can read it at:
http://www.numbersusa.com/interests/birthrightcitizenship.htm