The Citizens have allegedly entered the race to sign the 23-year-old France international amid uncertainty over the future of Chelsea target Nathan Ake.
Kounde, who is valued at £65m, is believed to be keen to seek pastures new and join a bigger club this summer having played over 130 games for Los Rojiblancos over the last three seasons.
According to the Daily Mail, Man City are closely monitoring Kounde's situation at Sevilla, but they remain behind Barca in the race for his signature.
Despite their financial concerns, the Catalan giants are currently finalising a deal to sign Leeds United attacker Raphinha, while they also planning a bid worth €50m (£42.3m) plus add-ons for Bayern Munich striker Robert Lewandowski.
Chelsea have also been linked with Kounde, although they have seemingly turned their attention to signing both Ake and Napoli's Kalidou Koulibaly, while they are also exploring the possibility of recruiting Presnel Kimpembe from Paris Saint-Germain.