The Uruguayan thanked a follower using the phrase “gracias negrito,” which translates to thank you, little Black one.
The phrase is seen as an affectionate term of endearment in South America but considered offensive in other parts of the world, because its meaning can be misconstrued.
Having come on as a substitute at halftime on Sunday with Manchester United 2-0 behind, Cavani had an instant impact, scoring twice and assisting the other goal to help his side to a dramatic victory.
Comparisons are being drawn with a 2011 incident involving another Uruguayan player, Luis Suárez.
Suarez, then a striker for Liverpool, repeatedly called then-Man United captain Patrice Evra “negro” during a Premier League game.
At the time, Suarez argued the term was not offensive in his homeland. The FA, however, disagreed with Suarez. A report into the incident said that Suarez did not use the term “negro” in “the conciliatory and friendly way that was common and inoffensive in Uruguay.”
Suarez was banned for eight games as a result and fined 40,000 pounds.