His agent, Robert Calenda, said he "reserves the right to take legal action" against the club after the video was posted and then deleted from the Serie A side's official TikTok on Tuesday.
Roberto Calenda released a statement on messaging platform X, formerly known as Twitter, saying the video caused "very serious damage to the player."
"What happened today on Napoli's official profile on the TikTok platform is not acceptable. A video mocking Victor was first made public and then, but now belatedly, deleted," said Calenda.
"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.
Nigeria international Osimhen scored 31 goals for Napoli last season as they ended their three-decade wait for another Serie A title and also reached the quarterfinals of the Champions League. Despite talks over a new contract opening months ago, the 24-year-old is yet to sign a new deal to replace his current one which runs until 2025.
Champions Napoli are seventh in the standings and play Udinese at home on Wednesday.