The 32-year-old flew into Manchester on Monday and completed a medical for a deal worth £6.5 million.
He told the club's official website: "I am very happy and really looking forward to the start of pre-season and performing well to help the team.
"This is a new challenge and a new league for me. I have been doing well in the previous years and my goal is to go on and hopefully perform at the same level for City.
"I know I am coming to a big club with a very good goalkeeper. City already have a great goalkeeper in Joe Hart but I will try and compete for the No.1 spot. I will do my best in the training sessions."
City have been seeking a new keeper to provie competition for Hart since Costel Pantilimon left for Sunderland earlier this summer.
He joins Bacary Sagna and Fernando in signing for Manuel Pellegrini's side this summer, with the acquisition of Brunu Zuculini soon to be confirmed.
"The most beautiful moments of my career were under Pellegrini in Malaga, where we achieved big things and we made Malaga's name known around Europe," Caballero added.
"As for my new team-mates, I have a very good relationship with Martin Demichelis but I've never met Pablo Zabaleta and Sergio Aguero. It's something I'm looking forward to. For me it is very good to have people from Argentina in the squad because I think it will be very helpful.
"I spoke to Roque Santa Cruz and Demichelis and they told me that Manchester is a very beautiful city and very welcoming so that will be great for my family and I can't wait to enjoy it.
"I've always loved the Premier League, I watched it as a child and I always liked the style of play here. Hopefully we can have many happy times together."