The Premier League champions made their move for the 27-year-old after seeing their overtures for Everton's John Stones repeatedly rebuffed.
Chelsea have conceded nine goals in four games in accruing four points to sit eight adrift of leaders Manchester City.
Chelsea were still to officially confirm the deal.
Ligue 1 club Nantes said in a statement on their website: "FC Nantes, Chelsea and Papy Djilobodji have reached an agreement for the transfer of the Senegalese international
Papy Djilobodji is a player of Blues for the next four seasons.
"FC Nantes extends its congratulations and wishes him well for the rest of his career."
Reports in the French media on Monday said a deal of around three million euros (£2.2m) for Djilobodji
It is less than 10 per cent of the sum offered for Stones.
Djilobodji had one year remaining on his Nantes contract and his likely switch to Jose Mourinho's Chelsea is unexpected.
It has been reported other Premier League clubs rejected the opportunity to sign him, but now he is playing for the defending champions and in the Champions League.
Source : PA
Source: PA