Maradona said Alves was "an idiot" and couldn't compare to other modern Brazil full-backs.
Canais Esporte Interativo (h/t ESPN) reported Alves' comments that started the back-and-forth:
"You want to compare Messi to Maradona? You can't compare them.
"I would not be proud to say that I won a World Cup with a 'Hand of God.' I wouldn't be able to tell my son that I won a World Cup like this with the whole world talking about it.
"The 'Hand of God' tricked us, you have to take this firm position. A sportsman like this can't be an example for youngsters."
As reported by TyC Sports (ht Football Italia), Maradona didn't take kindly to the comments, saying:
"Dani Alves is an idiot.
"Cafu and Maicon were real full-backs, he makes 28 crosses per game and only connects with four, poor thing.
"He speaks because he plays in a position on the pitch where football is not played.
"They touch the ball three times and make eight fouls per game.
"If we're going to keep talking about this idiot it's better that I go."
Maradona is widely regarded as one of the greatest players of all time and is perhaps most fondly remembered for leading Argentina to the 1986 World Cup. The team wasn't expected to compete in Mexico, but the then-Napoli star took the Albiceleste on his back and all the way to the title.
It was the player's finest moment, but it didn't come without controversy. In the quarter-finals against England, he bagged Argentina's first goal with a clear handball, later dubbed the "Hand of God."
Minutes later, he added a second with another iconic strike, this time after a long, mazy run. Argentina won the match 2-1 and would go on to beat West Germany in the final.
The Argentinians haven't won a World Cup since, a statistic Maradona's fans often use in debates regarding Messi's status as an all-time great.