The supporter was wearing a shirt with Vinicius Junior's name and number on the back at the Madrid derby, which Atletico won 3-1.
A young black girl, her mother and aunt were reportedly intimidated and the subject of abuse by Atletico fans.
Police have been alerted to the incident and the prosecutor's office has received an official complaint.
According to The Athletic, ex-Madrid striker Emilio Butragueno has invited the girl to the club's hospitality box for their La Liga fixture against Las Palmas on Wednesday evening.
Incidents of racist abuse from Atletico supporters towards Real players and their fans have been more frequent in recent meetings between the two rivals.
In September last year, Atletico fans made racist chants about Vinicius outside the stadium before the game. A section of supporters also made monkey noises throughout the game.
Atletico denounced the supporters on that occasion, saying the chants "provoke enormous repulsion and indignation. We will not allow any individual to hide behind our colours to utter insults of a racist or xenophobic nature".
The incidents did not stop there, however, as the Brazilian winger hit out at La Liga after he was targeted by Valencia fans.
"I'm so sorry. The championship that once belonged to Ronaldinho, Ronaldo, Cristiano [Ronaldo] and [Lionel] Messi today belongs to racists," he wrote before sharing a video compiling the many racist attacks he was the victim of throughout last season.