Napoli said they "never meant to offend or mock" Victor Osimhen in a social media post but stopped short of publicly apologising to him.
The club shared a video on their TikTok account of Nigeria's Osimhen missing a penalty, dubbed with a high-pitched voice saying "gimme penalty please".
The 24-year-old's agent has threatened legal action against the club.
A Napoli statement said it was "never the club's intent to offend Victor" should he "have perceived it that way".
The social media post has since been deleted.
"So as to avoid the topic being manipulated in any way, Napoli points out that it never meant to offend or mock Victor Osimhen, who is a technical asset of the club," a statement on the club's website read.
"Proof of this is the fact that the club firmly rejected all the offers it received for the attacker's transfer abroad."
Napoli boss Rudi Garcia said Osimhen is "invested 100%" in the Serie A club after he scored in a 4-1 win over Udinese on Wednesday - the first game since the incident.
Osimhen scored his side's second goal in a comfortable win but made no attempt to celebrate before he was mobbed by his team-mates.
Nigeria international Osimhen joined Napoli for a club-record 81.3m euros (£70m) in the summer of 2020 and he was integral to helping them win a first Serie A title in 33 years last season, scoring 26 goals in 32 league appearances.
"I can assure you that Victor loves this jersey," Garcia said. "He is invested 100% in our project.
"I have a very good relationship with Victor. I am happy for him because he scored tonight. After the match in Bologna, we were all frustrated, angry, Victor in particular, because he missed the penalty.
"On the sporting aspect and on other levels, especially on TikTok, there have been a lot of things to manage in recent days."
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