The Spanish striker had been linked with a move to Everton and Tottenham Hotspur earlier in the summer, but City are now ready to discuss personal terms with the player, according to Goal.com.
Sevilla accepted the bid after talks on Tuesday morning. Negredo has since left the club's pre-season training camp.
Should a deal be finalised, Negredo would become the second player after Jesus Navas to depart Sevilla for the Etihad.
Reports also surfaced yesterday that City had agreed a fee for Stevan Jovetic.
Negredo scored 25 goals in La Liga last season.