The 26-year-old centre-back, who has 12 caps, is expected to complete a £2.7m transfer today after the off-form Premier League champions failed to prise top target John Stones from Everton.
Djilobodji is in the last year of his contract at Nantes, who he joined in 2010, and had been training away from the first team amid speculation over his future.
He has made more than 180 appearances for the Ligue 1 side, although the majority came in the French second tier, and stands to cost a fraction of the outlay Chelsea were ready to sanction for Stones.
Everton rejected a fourth offer, believed to be for £38m, for the 21-year-old England star last week and turned down the player’s formal transfer request.
A home defeat to Crystal Palace on Saturday left Chelsea with just four points from their first four games of the new season.
West Ham, meanwhile, could clinch the return of midfielder Alex Song on a busy final day of the transfer window at Upton Park.
Cameroon international Song, who spent last season on loan at the Hammers, is expected to arrive on another temporary deal from Barcelona, while the club have had a £7m bid accepted for Nottingham Forest winger Michail Antonio and are also believed to be interested in Hull striker Nikica Jelavic.
Crystal Palace are set to offload forward Glenn Murray to newly-promoted Bournemouth for £5m.