Home fans could be heard making the chants during Nice's 4-2 win over Saint Etienne in Ligue 1 on Wednesday.
Nantes defeated Nice 1-0 in the Coupe de France final last Saturday.
Sala died in a tragic plane crash along with pilot David Ibbotson in January 2019 when traveling to England to complete a £15m transfer from Nantes to Cardiff City.
The light plane was found 67 metres underwater in February, 21 miles off the coast of Guernsey in the English Channel, with the search party retrieving Sala’s body but that of the pilot was never found.
A Nice statement read: "OGC Nice condemns in the strongest terms the chant about Emiliano Sala heard on Wednesday evening at the Allianz Riviera.
"The club does not recognise 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 extends its support to the family and loved ones of Emiliano Sala."
Nice manager Christophe Galtier, added: "I don't have words to describe what I heard."
"One of the first reactions in the dressing room was not our songs, shouts of joy or relief. It was to talk about what the players heard.
"On behalf of my dressing room, my technical staff, my medical staff, the players, we would like to apologise to the family of Emiliano Sala and FC Nantes."
A Nantes statement added: "FC Nantes has discovered with horror the content of the chants of some OGC Nice supporters and can only strongly condemn such acts 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."
