The club are believed to be in the market for a world-renowned talent following the exit of Wayne Rooney.
Modric has made 321 appearances in all competitions for Los Blancos since joining from Tottenham in 2012 - scoring 21 goals and assisting a further 52 during that time.
A stable in midfield for Madrid during the team's historic four Champions League wins in five seasons, the Croatian has seen his role reduced somewhat this season with the emergence of Federico Valverde.
DC United are in the process of revamping their squad following the departures of Rooney and Argentine attacking midfielder Luciano Acosta.
Ola Kamara has already signed from Chinese club Shenzen, while the team is reportedly also targeting Peru international Edison Flores (Morelia) and Uruguayan midfielder Brian Lozano (Santos Laguna).