Roberto Calenda released a statement on messaging platform X, saying the video caused "very serious damage to the player".
"We reserve the right to take legal action and any other useful initiative to protect Victor."
Napoli did not immediately respond to a request for comment outside normal business hours.
Osimhen missed a late penalty in a goalless draw against Bologna in Serie A on Sunday.
He was substituted as the game drew to a close and appeared to protest the decision, shouting at coach Rudi Garcia before leaving the pitch. The French manager, however, played down the situation.
Champions Napoli are seventh in the standings and play Udinese at home on Wednesday.