City boss Manuel Pellegrini is in the market for a new striker after selling Carlos Tevez to Juventus last month.
A whole host of names have been linked with a move to City, but it appears Negredo is emerging as their top target.
Negredo scored 25 goals for Sevilla last season and Sevilla are resigned to losing him after failing to qualify for Europe last season.
Atletico Madrid were thought to be leading the race for the 27-year-old, but now City are hoping to lure him to England.
If City sign Negredo he will join former Sevilla team-mate Jesus Navas in making the move to the Etihad Stadium this summer.