They certainly ARE, under current law. The thing for Congress to do (which it has refused to do so far) is to define "subject to the jurisdiction thereof" as excluding those in the country illegally. This would do an "end run" around the 14th, which was written to avoid declassifying blacks as "persons" and not to grant citizenship willy nilly to anyone here.....
However, in saying so, it is important to note that in the 19th century we had NONE of the ridiculous rules for entering this country we now have. The vast majority of the immigrant parents/grandparents of proud freepers who prate about "coming here legally" simply got on a boat with NO PAPERWORK AT ALL. Ellis Island (and scores of other immigrant sites) simply procesed them through, with papers or as WOPS (without papers, the origin of this Italian slur). The idea of passive eager immigrants lining up to get their "visa" outside the slovokian embassy is so idiotic as to be absurd.
Our immigrant problems are NOT the result of Mexican peasants who are lawbreaking goldbricks, eager to exploit our federal grabbag of giveaways. The problems stem from our welfare system and endless "programs" themselves. We are afraid to admit people who, even with unemployment tickling 7%, without whom large sections of our service industries would simply collapse. If we admitted them legally, they will be swarmed by "social" organizations pumping them to ride that gravy train. I have seen it happen.
Immigrations should be EASY to get here legally, and HARD to get here illegally. Once here legally, no "benefits" to non citizens. Of course, our federal gov't has the situation reversed. Natch.