The 19-year-old Spaniard has signed an initial two-year deal with the Championship side, with the option of a further year.
Rams manager Steve McClaren told the club's website: "After a successful trial period, we are delighted to welcome Ivan to the club.
"He is a young player with an excellent pedigree and we are really pleased that he sees his future at Derby County."
Calero spent the last two seasons playing for Atletico's B team and was involved with the first team in 2013.
Calero has appeared at various youth international levels for Spain and he has been on trial with Derby for the past two weeks, scoring in their pre-season friendly with Mickleover Sports.
McClaren added: "We are looking forward to working with him in order to help him develop at what is an important stage of his career."