The tweet – which was captioned: “Guess who?” – compared a photo of a young Benjamin Mendy, his City team-mate, with the mascot of Spanish confectionery brand Conguitos.
Mendy appeared to respond in good humour but the tweet was deemed offensive by many users of the social media platform and subsequently deleted.
Bernardo later posted a follow-up tweet, reading: “Can’t even joke with a friend these days... You guys...”
It is understood that the FA are aware of the tweet and will decide whether to take any action against Bernardo on Monday.
The 25-year-old scored a hat-trick in City’s 8-0 win over Watford on Saturday but could face a ban or fine if punished by the governing body.
Bernardo and Mendy are close friends, having played together at Monaco before both joining City in the summer of 2016.
Mendy did not appear to be offended by Bernardo’s original tweet, replying with smiley emojis and the message: “1-0 for you will see”.