The winger was stretchered off in Wednesday's LaLiga encounter, with Los Blancos saying he would be assessed by their medical staff in the coming days.
The Spanish champions have now provided an update, with the club confirming damage to a tendon in his hamstring.
"Following tests carried out on our player Rodrygo by the Real Madrid Medical Services, he has been diagnosed with a muscular injury affecting the tendon in his right biceps femoris hamstring muscle,” the statement on the club’s official website read.
Madrid did not provide any update on timeframe, but reports in Spain suggest that it is a serious injury that will rule him out for the bulk of the remainder of the season.
AS claims that the 19-year-old is not expected to return before the end of March, meaning he will only be available for the final weeks of the season.
Rodrygo posted a Christmas message to his followers on Instagram, with the picture showing him walking with the aid of crutches.
The news will be a huge blow to manager Zinedine Zidane, as his side start to find a run of decent form.
Madrid have won six on the spin in all competitions, and Zidane will hope the rest of the squad can make up for the absence of the forward.
“I'm thrilled for the players because it's not at all easy to win six games in a row,” Zidane said. “We deserved the win following our second-half performance.
“We have to be pleased because winning six in a row is no mean feat. So I'm extremely grateful to the players.”
Marco Asensio replaced Rodrygo against Granada, while Eden Hazard is also an option for Zidane as the Belgian works his way back to full fitness.
“We're going to rest a bit and then Eden will gradually come back into the team," Zidane said following the win over Granada.
Real face Elche in LaLiga on Wednesday and Hazard will hope to play some part as the team look to keep pace with Atletico who they are level on points with at the top of the table.