Navas has become the Blues' first summer signing after joining from Seville in a deal that could reach £24m.
European giants Real were also desperate to land the Spanish international, says the Daily Star.
Navas suffered from severe homesickness earlier in his career which often resulted in panic attacks.
But the Blues have been assured he is long over the problem and his arrival could now bring an end to Scott Sinclair's miserable 12-month spell at the Etihad Stadium.
Navas added: "It was my choice. I'm happy with the decision.
"I wanted to continue growing there and they put a lot of faith in me and I want to continue doing well there."