Savić was born in Mojkovac, which is a town in Montenegro. He started his career in football at the local club, Brskovo, before moving to a bigger team, BSK Borča. After playing with BSK Borča for 2 years, he moved to another football team, FK Partizan. While on Partizan, he won The Double, which is winning a country's top tier tournament and primary cup competition in a single season. After that, he was paid 6 million pounds to join Manchester City, winning the Premier League title in his only year playing on the team. In 2012, he was transferred to Fiorentina, where he played three years. He joined Atlético Madrid in 2015, where he won the UEFA Europa League and UEFA Super Cup.