Savić joined Atletico from Fiorentina for €10m in 2015, making 36 appearances for the club last season as they finished second in La Liga and won the Europa League final, for which Savic was an unused substitute.
Chelsea are seeking to bolster their defensive options after missing out on Champions League football. With David Luiz and Gary Cahill now in the twilight years of their careers, the Blues need reinforcements for the promising new pairing of Andreas Christensen and Antonio Rudiger.
Savic is known to Premier League fans from his one season at Manchester City, after Roberto Mancini brought him to England in 2011.
The Montenegrin international made just 11 league appearances and failed to impress though, and when Manuel Pellegrini was appointed at the end of the season, he was sold to Fiorentina in a swap deal with Matija Nastasic.
Atletico are getting a lot of their transfer business early, having already completed the acquisition of defensive midfielder Rodri from Villarreal last month.
Monaco’s Thomas Lemar is expected to complete a move to the Wanda Metropolitano in the next few days as well, after terms were agreed between the two clubs.
But they may lose star striker Antoine Griezmann, long linked with a switch to Barcelona. He has previously promised to announce a decision on his future before the World Cup, but time is fasting running out for him to do so.