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Cavani inspired United's comeback 3-2 victory over Southampton on Sunday with two goals after coming off the bench.
Replying to a message of congratulations on Instagram, Cavani used the Spanish term 'negrito', which translates as 'black'.
United stressed that the word was clearly used in an affectionate manner and has different connotations in South America, where Cavani is from.
The club said the 33-year-old had been made aware such terms are viewed as offensive in the UK and he subsequently deleted the post - though the FA is aware of the matter and will investigate.
On Monday afternoon, Cavani released a statement which read: “The message I posted after the match on Sunday was intended as an affectionate greeting to a friend, thanking him for his congratulations after the game.
"The last thing I wanted to do was cause offence to anyone. I am completely opposed to racism and deleted the message as soon as it was explained that it can be interpreted differently. I would like to sincerely apologise for this.”
United added: “It is clear to us that there was absolutely no malicious intent behind Edinson’s message and he deleted it as soon as he was informed that it could be misconstrued.
"Edinson has issued an apology for any unintentional offence caused. Manchester United and all of our players are fully committed to the fight against racism.”
The FA issued guidelines ahead of this season clamping down on racist and discriminatory language and behaviour, with offences on social media carrying a minimum three-game ban.
Manchester City's Bernardo Silva and Dele Alli of Tottenham were both suspended for one match last season after posts that breached FA guidelines.
Anti-discrimination group Kick It Out had earlier released a statement regarding Cavani's initial post, which read: "We are aware of a comment made last night on social media by Edinson Cavani that, in its native tongue, can be offensive in some contexts.
"We understand that the FA is investigating and the club and player are co-operating with that investigation. We await the outcome of that investigation.
"As we have said before, we believe that it would be helpful for overseas players, coming to England, to have routine education on language that may be unacceptable in this country."
Additional reporting by PA.