The Spanish winger will sign for City for a fee reportedly around the £15 million mark.
Navas told the City website: "Manchester City is an exciting project and this is the right moment for me to take this step. I'm very happy with the opportunity and the decision.
"The Premier League is a competition where the football is very fast and will suit the way I play. I'm really looking forward to it. City have been successful in recent years and are continuing to make a great team and I'm honoured to now be part of it."