The 55-year-old, who has previously managed Mexico, has been in charge of the Spanish outfit since November 2012.
The experienced manager joined when the club were bottom of La Liga, but secured safety and has led the Barcelona-based side into 13th spot this season.
However, Aguirre will now seek pastures new, with a return to international management on the cards.
"We've worked comfortably at all times, but both the club and myself decided it was for the best and that I wasn't going to listen to the offer they were going to make," Inside Spanish Football quotes Aguirre as saying.
"There's been no trauma or argument, it's just the end of a cycle. The history of the club comes before any individual and now, I'll walk a different path."
Espanyol lost 18 of their 38 league games this season.