The incident occurred as Vinicius made his way around the Camp Nou pitch after being substituted in the 87th minute of the 2-1 victory for Los Blancos on Monday (AEDT).
LaLiga has decided to report the incident to the Prosecutor's Office in Barcelona in case it constitutes a hate crime.
A statement from LaLiga read: "Following the events that occurred in the match held at Camp Nou last Sunday, in which a home fan abused Real Madrid player Vinicius Jr with racist insults, LaLiga will proceed to report the event to the Hate Crimes Chamber of the Prosecutor's Office in Barcelona in case they constitute the crime outlined in Article 510 of the Penal Code.
"LaLiga, as it did when [Athletic Bilbao's] Inaki Williams received racist insults at the home of Espanyol in January 2020, will appear as a private complainant to cooperate in the fight against any type of act or behaviour that is racist or xenophobic.
"LaLiga has been fighting this kind of behaviour for years and promoting the positive values of sport."