The 30-year-old only joined Milan on a permanent contract in July after spending last season on loan at the San Siro from Fiorentina.
Kalinic was sent home from the World Cup reportedly for refusing to come on as a substitute in Croatia's opening game against Nigeria, though coach Zlatko Dalic said the player had a back injury. Croatia went on to reach the final.
The former Blackburn and Dnipro forward scored six goals in 41 appearances for Milan last term and has netted 15 times in 42 international matches.
Kalinic will provide back-up to French World Cup winner Antoine Griezmann, Spain star Diego Costa and Frenchman Kevin Gameiro at the Wanda Metropolitano.
Atletico get their Spanish La Liga season underway with a trip to Valencia on Monday, August 20.