The pair were caught on camera exchanging words throughout the encounter played in the Israeli city of Tel Aviv.
Uruguay captain Diego Godin at one point stepped in to intervene during one of the heated exchanges in the first half.
Before half-time, after Messi had been fouled by Matias Vecino, Cavani reportedly told the Barcelona forward to "fight it out," to which Argentina's No. 10 replied: "Whenever you want."
The argument between the two reportedly continued in the tunnel at half-time with both players trading insults.
Cavani, who scored Uruguay's opener and was substituted before the hour, was asked about the bust-up with Messi and told TyC Sports after the game: "It's part of football, Clasicos are like this.
"The truth is that it was at times a game that had friction, there were challenges and things that are part of football, especially of South American [football].
"We take positives from our performance and the fact that we were able to score two goals. We still need to improve but it's good to end the year like this."
Messi scored a late penalty to ensure Argentina ended the year with a 2-2 draw.
Argentina will be without Messi when they begin their South American qualifying campaign for the 2022 FIFA World Cup Qatar at home to Ecuador on March 26.
Messi must serve a one-match ban for his red card against Chile in the third-place game of the 2019 Copa America.