The La Liga club rejected an initial €55 million City offer for centre-back Kounde last week, believing they'll be able to sell him for more next summer.
However, sources told ESPN that if City were to raise their bid to €65 or €70m, Sevilla would find it extremely difficult to refuse in the challenging, post COVID-19 financial environment.
Sevilla's sporting director Monchi signed Kounde, 21, as a relative unknown from Bordeaux last summer for €25m.
He went on to become one of the most impressive central defenders in La Liga, forming a Europa League-winning partnership with the Brazilian Diego Carlos.
Sources close to the club told ESPN that they feared Thursday night's UEFA Supercup defeat to Bayern Munich could be Kounde's last game in a Sevilla shirt.
The France U21 international's release clause exceeds the fees being discussed -- but there is an internal debate at Sevilla as to whether they would be better off waiting for his value to increase before selling next summer, or accepting an improved offer from City now.
ESPN reported in July that Real Madrid were also tracking Kounde, with an eye to moving for him in 2021.
Kounde's initial plan was to remain at Sevilla for another season, but he would be tempted by a move to a club like City now, sources say.
He also believes he would have more opportunities to play regular football in Manchester than at other clubs of a similar stature.
Pep Guardiola has already strengthened his defence with the signing of Nathan Ake from Bournemouth this summer but is keen to bring in another centre-back.
Atletico Madrid's Jose Maria Gimenez, Napoli's Kalidou Koulibaly and Villarreal's Pau Torres have all been looked at, but Gimenez and Koulibaly are thought to be too expensive, and Torres is a longer-term prospect.