The 31-year-old made 284 appearances for Arsenal after joining them from Auxerre in 2007. His contract expires on June 30.
He has won 41 caps for France who face Switzerland, Ecuador and Honduras in Group E of the World Cup.
France kick-off their campaign against Honduras in Porto Alegre on Sunday.
Sagna, who is being given the squad number three, is expected to join City's squad in time to join them on a pre-season tour to the United States. He wore the same squad number with Arsenal.
He is the first major addition to his triumphant squad made by City manager Manuel Pellegrini.
Commenting on Twitter, Sagna said: "Of course I wanna [sic] thank the coach Mr Wenger, who believed in me, gave me trust and clearly changed my life."
He described Arsenal as "A family to me... a club who taught me so much in seven years. A club who changed the kid I was into the man I am today."
(Reporting by Tim Collings, editing by Pritha Sarkar)
