The midfielder was substituted at half-time during Tuesday's Champions League meeting with Aston Villa after experiencing discomfort in his right leg. Tests carried out on Wednesday confirmed a small injury to the biceps femoris muscle, ruling him out of the trip to the Metropolitano.
The club have not given a specific recovery date, with his return dependent on how the injury develops.
"Rodrigo Hernández has undergone medical tests following the muscle discomfort he felt during yesterday's match against Aston Villa," Barcelona said in a statement.
"The tests confirm that he has suffered a minor injury to the biceps femoris muscle in his right leg. He will miss next Sunday's match against Atletico Madrid. The exact recovery time will depend on the development of the injury."
Hansi Flick had indicated after the Aston Villa game that the problem did not initially appear serious.
"I substituted Rodri because he had a little discomfort," the Barcelona coach explained.
The injury brings an end to a lengthy run of appearances for the 30-year-old, who made his Barcelona debut against Athletic Club on August 27. He has featured in all of the club's matches in LaLiga and the Champions League since then, accumulating 945 minutes across 14 appearances.
Rodri has started 11 of those games and become an important part of Flick's midfield partnership with Pedri. His ability to recover possession, control the tempo and dictate play has made him a regular selection, with the coach having limited opportunities to rest him.
His absence against Atletico is therefore a significant setback, while the timing also raises doubts over his availability for Barcelona's first fixtures after the international break.
The club already have several players working away from the main group, including Roony Bardghji, Andreas Christensen, Dani Olmo, Eric García, Jules Koundé and Xavi Espart.
Barcelona will now monitor Rodri's recovery before determining when he can return to action.