Dutchman Drenthe is officially owned by Real Madrid, and came up against Messi during his time in Spain.
Uruguayan Suarez was banned for eight matches and fined £40,000 earlier this season after being found guilty of using abusive and/or insulting words and/or behaviour in a Premier League match in 2011.
In an article on Dutch news website Nu.nl, Drenthe says: "I played against him many times and we always have problems with each other.
"You know what bothers me so? That tone with which he always says, 'negro, negro'. I understand that 'negro' in South America is very common, but we can not stand it.
"Mahamadou Diarra, my team-mate at Real, could explode if 'negro' was aimed at him. The Argentinean Gabriel Heinze and Gonzalo Higuain said it initially on the training ground, but they were stopped.
"When Hercules (the team Drenthe spent a year on loan with last season) played Barcelona, during the game I had a small altercation with him. He gave me a hand in the match and again said a few times, 'hola negro'."