Silva is the third summer signing for City with the club having already lured defenders Jerome Boateng and Yaya Toure.
The Spaniard will, however, not join up with his new team-mates until after he returns from a holiday to the Canary Islands.
That will be Silva's first chance to relax since helping Spain to their World Cup win.
Silva was only briefly used during the tournament, making just two appearances after starting in Spain's opening defeat to Switzerland.
The winger was not used at all in the 1-0 final victory against Holland on Sunday and Silva admitted he hoped his club form with City this season could help him push for a regular starting role with Spain.
"I'm very happy with what we did this summer - it's the best time that any player can have," he said.
"Of course, if you play well for your team and things go well, the national coach is going to see that and you become more important to him."