The report claims that the 27-year-old defender has reached an agreement with the Blues and will under a medical.
Djilobodji has a year remaining on his current deal with Nantes and has nine goals as well as two assists in 183 appearances for the Ligue 1 outfit since 2010.
Chelsea manager Jose Mourinho is reportedly concerned with the state of his sides defending after conceding nine times in their first four league games.
Trabzonspor also made a £1.4 million bid for Nantes target Djilobodji according to source close to the Ligue 1 club.
The Mirror report that Premier League outfit Sunderland are also interested in the 27-year-old defender.
Trabzonspor have had a busy transfer window and have already signed Cavanda from Lazio, N’Doye from Hull City, Europa League winner M’Bia and Turkish international Okay Yokuşlu.
The Black Sea Storm drew 2-2 with Akhisar at the Avni Aker stadium to remain in first place.