The Istanbul giants have been monitoring the Senegalese defender for the past three months and are preparing to make a move for the 26-year-old.
Nantes have valued the powerful defender at €6,5 million however, Fenerbahce are looking to negotiate a lower price.
The Sunday People have stated that Southampton have opened negotiations with Nantes over the signing of Papy Djilobodji.
Bournemouth – who have just been promoted to the Premier League – are also said to be on the the Senegalese international who has a year remaining on his contract with FC Nantes.
Nantes are currently in 13th place in the French Ligue 1 with 45 points in 37 games. Les Canaris take on Lens in their final game of the season.
Papy Djilobodji made 32 appearances for FC Nantes in all competitions this season.