Hello, I am an American citizen living in Madrid. We came here with a program to get our TEFL certificate. We asked the program if we needed a visa and they said no. We are here now, we have a place to live, and start the program soon. However everything we try to set up requires an NIE. Is there any way to get this if we didn’t come to Spain with a visa? Thanks for any help

atlenmadrid,
you are practically tourists. supposed to be touring. not studying.
once you overstay you are practically illegal aliens.
you will need to consult a lawyer.
in any case, after reading page two of the ff

http://www.mir.es/SGACAVT/modelos/extranjeria/modelos_extranje/ex_14.pdf

it sounds simple. try it before you become illegal aliens. and consult a lawyer.