The 14-second video - which was hastily deleted - shows Osimhen appealing for a penalty in the weekend's Serie A encounter with Bologna, with a high-pitched voice in the background squealing "give me penalty please" on two occasions.
The clip then cuts to referee Giovanni Ayroldi awarding the spot kick for handball, leading to the same squeaky voice saying: "Thank you, you give me penalty. Thanks!"
Osimhen stepped up to take the penalty at the Stadio Renato Dall'Ara but fired wide of the post, at which point the video seemingly mocked the attacker's run-up before a final inaudible comment as the ball sails out of play.
Osimhen's agent Roberto Calenda subsequently released a statement blasting Napoli's now-deleted post, claiming that the clip caused "very serious damage" to his client, and legal options are now being explored.
"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," a statement read.
"A serious fact that causes very serious damage to the player and adds to the treatment that the boy is suffering in the last period between media trials and fake news. We reserve the right to take legal action and any useful initiative to protect Victor."
Alongside the bizarre penalty video, a second clip allegedly posted from the Napoli TikTok account portrayed Osimhen as a "coconut" with an upbeat song playing in the background.
In response, Osimhen appears to have deleted most Napoli-related posts from his Instagram account, and the former Lille man also came to blows with head coach Rudi Garcia after being substituted towards the end of that draw.
By blasting his penalty wide, Osimhen stretched his goalless run in all tournaments to four matches, and Garcia withdrew the 24-year-old for Giovanni Simeone with 86 minutes gone at Dall'Ara.
Upon exiting the field, Osimhen and Garcia engaged in a war of words, but the Partenopei coach refused to reveal what was said between the pair behind the scenes, telling the media: "Even the greatest players in the history of football miss penalties. As for when he went off, we talked about that, what I say to my players remains in the locker room.
"We have a lot of quality in the squad and on the bench, so Simeone, [Matteo] Politano and [Eljif] Elmas came on, I still have Jesper [Lindstrom] available too. When we have this much quality, we can't always have 90 minutes with the same men, especially as we have four games in nine days."
Osimhen inspired Napoli to their first Scudetto triumph in 33 years last season with a league-high 26 goals in 32 matches, as well as five goals in six appearances in the Champions League.
The Nigeria international was heavily linked with a move to the likes of Chelsea, Manchester United or Paris Saint-Germain in the summer, and top European clubs are expected to monitor the latest developments closely with a view to a possible transfer in 2024.
Osimhen remains under contract at Napoli until the summer of 2025 and has amassed 62 goals and 15 assists in 107 games for the club since joining from Lille in 2020.