Reports in Spain claim that Ramos was impressed by what he saw of Odriozola on international duty with Spain and believes that he would be a good addition to Zinedine Zidane's squad.
The 21-year-old made his international debut during the domestic break, starting the 3-0 World Cup qualifying victory against Albania in the absence of Dani Carvajal and getting an assist in the process.
Spanish publication 'Diario Gol' reports that Ramos liked what he saw from the Sociedad star and has since told Real's club president Florentino Perez to do a deal for the player.
The youngster has been on Real's radar for some time now and he was one of the options that the club considered to replace Danilo after he joined Manchester City in the summer but it now seems that 'Los Blancos' may pursue a deal in January.
Perez and Real will have to be quick if they plan on trying to secure the player's sevices though, with Barcelona also thought to be admirors of the right-back.
Odriozola is under contract in San Sebastian until 2022, though his contract does include a €40m release fee that Real may have to activate if they are to get their man.