JJ is correct. In the theme with the taxi horns-sounds of Paris, there are a couple of 4-5 note excerpts from popular marches (just enough to allow recognition), often heard at that time in France. These by no means constitute a "theme." "An American in Paris" was a strikingly original piece at its time, and is a landmark in composition.

Gershwin went to a parts store in Paris and bought auto horns so that he could properly transcribe their sound to instruments in the orchestra.   

"Sorella-Maxixe-Mattchiche," sheet music generally sold under the last name, is still a popular march, particularly in Latin America, and, along with "Zacatecas," is played ad nauseum. One's brain becomes infected with these tunes. Sheet music is readily available for bands in a number of editions.

If there was a "traditional Spanish song" as a predecessor, I have been unable to find lyrics. It may have been a dance tune, or a compound of several tunes.