The Argentinian goalkeeper joined the Barclays Premier League champions this week, in a £4.4m move from Spanish club Malaga.
The 32-year-old will replace released Romanian Costel Pantilimon as the main back-up for Joe Hart, but hopes to compete with the England keeper for a place in the first team.
Caballero said: "I knew that Manchester City were interested in me so I tried to do my best for my previous club to make the move to a very big club possible, and finally it's a reality and I will try to prove myself.
"The first goal is to do things well and show why City were interested in me.”
Caballero won the U20 World Cup with Argentina in 2001 and was also a member of the 2004 Olympic gold medal-winning squad.
"Obviously it would be good to get back with the national team and hopefully I can achieve that,” he said.
This is the second time that Chilean manager Manuel Pellegrini has signed the keeper.
Pellegrini signed the former Boca Juniors keeper for Malaga from Elche in 2011, and returning to his management is part of the appeal of the Etihad for Caballero.
Caballero, who is City's third summer signing after Bacary Sagna and Fernando, said: "I think Pellegrini’s best quality is giving players the confidence to achieve their potential."