Konate is set to join Real on a four-year contract after leaving Liverpool and will be officially confirmed once Florentino Perez is elected.
The Frenchman will be closely followed through the Santiago Bernabeu doors by Inter Milan wing-back Denzel Dumfries after the Spanish giants triggered his €20m release clause.
Los Blancos have bolstered their back line significantly by agreeing the two defensive signings, and according to Fabrizio Romano, Jose Mourinho has "approved" the deals for Madrid.
Mourinho will officially complete his return to the Spanish capital, 13 years after leaving the club, when the presidential election is finalised on June 7.
The Portuguese tactician has agreed a two-year contract at the Bernabeu that includes an automatic one-year extension if he wins the LaLiga title.
Real Madrid watched rivals Barcelona collect their second successive league title last month, but Mourinho will hope to transform their fortunes, starting with the additions of Konate and Dumfries.