Manager Pep Guardiola is planning to be active in the up coming transfer market and space needed to be made for wages.
Sagna joined the club from Arsenal in 2014 and went on to make 85 appearances for the Citizens.
Clichy, who also joined from Arsenal, has made more than 200 appearances for the club and was apart of the side that won City's first title in 44 years back in 2012.
Spain international Navas moved to England from Sevilla in 2013. The Spaniard started 117 games for club, picking up a Premier League winner's medal in 2014 and two league cups in his stint.
Caballero became cemented his legacy at the club last season after starring in the penalty shootout against Liverpool in the League Cup final. The goalkeeper made 27 starts under Guardiola this season, and the Argentine expressed his gratefulness for the time he had with the Citizens.
Caballero told the club's website: "It's been a privilege to play for Manchester City.
"From the first day I arrived, everyone at the club has made me feel welcome and I will always be grateful to them.
"To win the League Cup in such dramatic circumstances was special and it's a day I will never forget."