Rodwell's move was announced just hours after Roberto Mancini's City troops defeated Chelsea in the Community Shield.
"I relish the opportunity of playing with some of the best players in the world and continuing my development," said Rodwell before heading off to meet up with the England squad for the friendly against Italy.
"It will be an amazing experience to make my debut in front of them if I am selected to play next weekend.
"The hard work starts today as I start to help contribute to the club winning more silverware. The season has started really well with the Community Shield win."
He added: "I would like to place on record my thanks to everyone at Everton Football Club. The Manager David Moyes has been brilliant for me and helped make me the player that I am today.
"I have a special relationship with the chairman, Mr Kenwright, who did not want me to leave but this is too good an opportunity for me to refuse."
Mancini, who had been frustrated at a lack of summer signings, had angered Everton fans after defeating Chelsea in the Community Shield by saying Rodwell was not yet ready for City's first team and that the player was now joining a club with a winning mentality.
He said: "Jack is a good player. He is young. It is the first time he is in a club that plays to win and it will be different for him.
"But in a couple of years I am sure he will be a top and stronger midfielder. He has good quality.
"Rodwell is a player that needs to improve to play in this XI but he is young and a good player for our future.
"We are interested in all good players. I liked him last season. For this reason, we bought him."
