The video that was posted and then deleted from Napoli's official TikTok on Tuesday and was followed by Osimhen deleting almost all pictures of himself in the team's shirt from his social media accounts and his agent saying he was considering taking legal action.
"To avoid any potential exploitation of the matter, SSC Napoli would like to explicitly state that the club never intended to offend or make fun of Victor Osimhen, who is an asset to the club," Napoli said in a statement on Thursday.
"Over the course of the summer Napoli rejected all offers to sign the striker -- firm proof of the club's appreciation of him.
"On social media, and TikTok particularly, expressive language is used in a light-hearted and playful manner. In this case involving Victor there was no intention of mockery or derision.
"If Victor was in any way offended, this was not at all in the club's intentions."
Speaking after Wednesday's 4-1 Serie A win over Udinese, in which Osimhen scored, Napoli head coach Rudi Garcia called the post "clumsy" but added the player is committed to playing for the Italian champions.
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 agreement which runs until 2025.
"I can assure you he loves this jersey and that he will give his all this year," Garcia said in a postmatch interview with DAZN.
"Nobody wanted to do harm, neither TikTok nor Victor on their social networks. No harm was intended but there were instinctive reactions. I don't know if now Osimhen will publish his photos on social media again with the [Napoli] shirt."
There appeared to be tension between Garcia and Osimhen during Sunday's draw in Bologna, with the forward looking visibly upset after being substituted. However, Garcia praised Osimhen's attitude in Wednesday's game.
"In Bologna, we were all frustrated and angry for not having won," Garcia said. "Victor in particular because he missed the penalty. We clarified things and everything returned to sporting normality. Then there were two days to manage due to a clumsy episode [the posting of the video]."
Osimhen scored Napoli's second goal against Udinese but did not celebrate as his teammates ran over to congratulate him.
He had earlier let teammate Piotr Zielinski take a penalty that broke the deadlock.
"I complimented him," Garcia said of Osimhen, who is Napoli's designated penalty taker. "I've had an excellent relationship with Victor since I arrived and I'm happy because he scored, a centre-forward always wants to score goals."