Videos emerged on social media of Barcelona fans racially abusing the Brazilian forward and it has been alleged that a banana was thrown at the Real Madrid star.
The Spanish top-flight has moved quickly to investigate the matter and is working with police to identify the culprits.
"La Liga has been working from the outset to identify the individuals involved in making racist insults and will file the relevant charges as is always the case," a statement read.
"We are coordinating with the Mossos d'Esquadra and FC Barcelona."
Meanwhile, Barcelona have announced that they are also investigating the issue, writing on social media: "FC Barcelona will always defend the values of football and sport such as respect for the opponent and we will investigate any racist insult that may have occurred this afternoon during the match against Real Madrid."
Vinicius, 23, has been a frequent target for racist abuse since arriving in Spain.
At least nine complaints about racial abuse directed at Vinicius were made to a court in Barcelona by La Liga last season.
The most high-profile incident came during a match against Valencia in May, resulting in Vinicius declaring on social media that La Liga "belongs to racists".
That comment sparked an angry reaction from La Liga president Javier Tebas, who posted: "Before criticising and insulting La Liga, you need to inform yourself properly Vinicius."
More recently, Vinicius vowed he will continue his fight against racism in Spain, saying: "I personally know that I am not going to change history, that I am not going to make Spain a country without racists, nor the whole world.
"But I know that I can change some things. So that those who come in the next few years do not go through this, so that children can have peace of mind in the future. For them I will do everything I can."