According to Spanish publication AS via the Sun, Magpies boss Rafa Benitez has made the 31-year old his top priority this summer.
Navas was released by City after his contract expired at the end of last season.
Speculation has strongly linked the Euro12 winner with a return to his former club, Sevilla.
However, the report suggest that Newcastle have ramped up their interest as Benitez looks to add players with Premier League experience to his squad.
Navas made 183 appearances for City and won the Premier League title in 2014.