"A tear in his left thigh has been diagnosed as a result of the blow suffered in the match between Colombia and Peru," Madrid said in a statement on their website.
The club refused to estimate how long the top scorer at the 2014 World Cup will be out of action, but he didn't train with the rest of his teammates on Thursday and seems certain to miss Saturday's trip to Espanyol.
Rodriguez was forced off with the injury midway through the second-half of Colombia's 1-1 friendly draw with Peru on Tuesday.
The 24-year-old had surprisingly been left on the bench by Rafael Benitez for Madrid's 0-0 draw away to Sporting Gijon to open their campaign due to his participation in the Copa America.
However, he responded in style in his first start of the season with two incredible goals in a 5-0 thrashing of Real Betis before the international break.
The Spanish giants are also set to be without right-back Danilo for up to a month after he was diagnosed with plantar fasciitis in his right foot.
The 24-year-old has made just two appearances for Los Blancos since joining from Porto in the close season for 31.5 million euros ($35.2 million, £22.9 million).
Reports in the Spanish media suggest the Brazilian international will miss up to four weeks, ruling him out of Real's Champions League opener against Shakhtar Donetsk on Tuesday and the Madrid derby away to Atletico in La Liga on October 4.