Nope!
Senator Jacob Howard, who, in 1866 fully described the intent, focus, and limits of the first clause while addressing and introducing it to the 39th Congress for ratification:
“Every person born within the limits of the United States, and subject to their jurisdiction, is, by virtue of natural law and national law, a citizen of the United States. 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. It settles the great question of citizenship and removes all doubt as to what persons are or are not citizens of the United States. This has long been a great issue in the jurisprudence and legislation of this country.”
Senator Jacob Howard, Congressional Globe, 39th Congress (1866) pg. 2890
By the senator's logic, those not born to citizens do not qualify. At the time of her birth, for example, NEITHER of Harris' parents were US Citizens, which makes her parents foreigners when she was born.