The two sports papers said that the transfer fee will be around 28 million euros (36.4 million dollars). There has been no official announcement from the two clubs.
"Manchester City is a club that I like a lot. I am open to any possibilities and I want to win trophies, that is what I want to fight for," Navas said after Sevilla's 4-3 defeat of Valencia Saturday.
Navas, 27, who helped Spain to win the World Cup in 2010 and Euro 2012, is in the Spain squad for the upcoming Confederations Cup in Brazil.
Another player about to leave Sevilla is veteran goalkeeper Andres Palop who is poised to sign a one-year contract with Bundesliga club Bayer Leverkusen. Palop, 39, helped Sevilla to win the old UEFA Cup in 2006 and 2007, and the Spanish cup in 2010.