A section of Nice fans could be heard singing offensive and distasteful songs about former Nantes striker Sala during Nantes' 2-1 home league win.
Sala was a passenger in a plane that crashed in the English channel in January 2019 while en route to Cardiff from Nantes, having just days earlier signed for then-Premier League side Cardiff City.
The club said in a statement: "FC Nantes have discovered with horror the tenor of the songs of certain supporters of OGC Nice and can only strongly condemn such actions degrading the memory of Emiliano Sala.
"At the same time, FC Nantes would like to thank OGC Nice and its coach Christophe Galtier for their support and their firm condemnation of such remarks. FC Nantes as a whole supports Emiliano's family and loved ones."
Nice said in a statement: "OGC Nice condemns in the strongest terms the song about Emiliano Sala heard on Wednesday evening at the Allianz Riviera.
"The club does not recognize its values or those of the entire red and black family in this unthinkable and abject provocation of a minority of its supporters.
"OGC Nice sends its support to the family and loved ones of Emiliano Sala."
Nantes boss Antoine Kombouare said after the game he wants those responsible to be banned.
"Human bulls--t has no limit," Komouare said. "I am scandalised, these people have nothing to do in stadiums. They should be banned, it's shameful, I feel sorry for the family. I knew that the supporters could be idiots, but these are donkeys."