The Belgian, who is currently on loan from Chelsea, was featuring for the Italian giants in their Coppa Italia clash against Juventus where fans were shouting racial slurs throughout.
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Juan Cuadrado gave the home side the lead, before Inter were awarded a penalty late in the game, with Lukaku stepping up.
M chants could be heard from some sections of the Juventus fans, as well as the Italian equivalent chants of 'f*** off' and 'f***ing m'.
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Taking the penalty, Lukaku converted past Mattia Perin before celebrating in front of the Juve fans with his finger over his lips, as well as being reported to say 'muto', which means 'shut up' in Italian.
But already on a booking, Lukaku was given a second yellow card for his celebration, something that has outraged Roc Nation president Michael Yormack.
In a statement, he said: "Tonight's racist remarks made towards Romelu Lukaku by Juventus fans in Turin were beyond despicable and cannot be accepted.
"Romelu scored a penalty late in the game. Before, during, and after the penalty, he was subjected to hostile and disgusting racist abuse.
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"Romelu celebrated in the same manner he has previously celebrated goals. The referee's response was to award a yellow card to Romelu.
"Romelu deserves an apology from Juventus, and I expect the League to condemn the behaviour of this group of Juventus supporters immediately.
"The Italian authorities must use this opportunity to tackle racism, rather than punish the victim of the abuse.
"I am certain that the footballing world shares the same sentiment."