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The 33-year-old Uruguayan striker sent an Instagram post thanking a friend for his congratulations after scoring twice as United came from 2-0 down to win 3-2 on Sunday that included the Spanish word "negrito" (small black person).
United said the word was clearly used in an affectionate manner and has different connotations in South America.
The former Paris Saint Germain forward had been informed such terms are viewed as derogatory in Britain and he subsequently deleted the post.
"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," Cavani said in a statement.
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"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."
The FA issued guidelines pre-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 the guidelines.
"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," United added in a club statement.
"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."
Cavani will be free to play against his former club PSG in the Champions League on Wednesday.
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