Kounde, 21, enjoyed an impressive debut campaign at Sevilla after joining from Bordeaux last summer for a club record €27.5 million.
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The defender made 29 La Liga appearances this term and helped Sevilla to qualify for next season's Champions League.
Madrid coach Zinedine Zidane was interested in Kounde last season and although club captain Sergio Ramos and Raphael Varane are undisputed starters at centre-back, Zidane wants more options for the upcoming campaign.
Kounde has formed one of the most solid defensive partnerships in Europe alongside Diego Carlos. Zidane likes Kounde's speed and good positioning on the pitch.
Sources have told ESPN that Madrid are also impressed with Carlos' physique and power but Zidane prefers to sign Kounde due to his youth and room for improvement.
Sources close to the player have told ESPN that several big clubs in Europe are monitoring Kounde which is why Madrid want to get ahead of their rivals for his signature.
Madrid president Florentino Perez said the club has no intention of making a big summer signing due to the coronavirus pandemic, but the club's sporting area is continuously working and following several future prospects.
Signing Kounde, who is tied to Sevilla until June 2024, to the Bernabeu would require a big investment.
The France under-21 international is valued at €35m but Madrid know that figure will continue to increase and, for the time being, the club are only monitoring him to see if his rise continues.
ESPN's Adriana Garcia also contributed to this report.