Azpilicueta had been a mainstay at Chelsea since he arrived from Ligue 1 side Olympique de Marseille in 2012, making more than 500 appearances in all competitions, scoring 17 goals, providing 56 assists and winning every major title with the club.
"Cesar leaves an everlasting mark at Chelsea, as a warrior, as a champion, and as loyal Chelsea legend," Chelsea chairman Todd Boehly and co-controlling owner Behdad Eghbali said in a joint statement.
"He has set the standards at the club for more than a decade and demonstrated to everyone what is required day in, day out to achieve success."
The Spanish defender is expected to join LaLiga side Atletico Madrid as a free agent after Chelsea agreed to let him go despite there being 12 months left on his contract.
The Champions League, Premier League, Europa League, FA Cup and League Cup winner was linked with a move to Serie A side Inter Milan before deciding on Atletico as his next destination.
He joins the growing list of players to leave Stamford Bridge this close season after the exits of Kalidou Koulibaly, N'Golo Kante and Edouard Mendy, who all joined Saudi Pro League clubs.
Kai Havertz, Mateo Kovacic and academy product Mason Mount have signed for rivals Arsenal, Manchester City and Manchester United, respectively.
Chelsea have taken on new players, with forwards Christopher Nkunku and Nicolas Jackson joining the club, as they seek to revive their fortunes under former Tottenham Hotspur manager Mauricio Pochettino after finishing 12th - their worst finish to a season since 1994.
