The Colombia international required surgery after damaging the fifth metatarsal in his right foot during Wednesday's 2-1 La Liga victory over Sevilla, in which he scored the opener.
Rodriguez is now set for a lengthy spell on the sidelines, but the forward remains in positive spirits.
"I will recover and come back stronger than ever," he posted on Twitter.
"The messages [I have received] are very supportive. Thank you!"
Rodriguez moved to the Santiago Bernabeu in a high-profile move from Monaco among much fanfare in the close-season, after impressing at last season's World Cup where he collected the Golden Boot award.
The former Porto star has scored 12 goals in all competitions for Real this season.
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