The Premier League champions have conceded nine goals in four games in accruing four points to sit eight adrift of leaders Manchester City.
Chelsea’s pursuit of John Stones has been repelled by Everton and reports in the French media say a €3m (£2.2) deal for Djilobodji is near completion.
It is less than 10% of the sum offered for Stones.
The 27-year-old Djilobodji has one year remaining on his contract with Nantes and has flown to London for a medical, according to reports in France.
His likely switch to Jose Mourinho’s Chelsea is unexpected.
The Blues have signed goalkeeper Asmir Begovic, left-back Baba Rahman and striker Radamel Falcao on loan from Monaco — all like-for-like replacements for the departed Petr Cech, Filipe Luis and Didier Drogba.
Pedro was added from Barcelona, boosting morale and the performance in a win at West Brom before Saturday’s loss to Crystal Palace.
Rumoured interest in Paul Pogba of Juventus refuses to subside, but it would take an astronomical fee to land the Frenchman, and Chelsea will require a U-turn from Everton to return for Stones.
Mourinho insisted Chelsea, who have allowed forward Lucas Piazon join Reading on a season-long loan, would not panic in the closing hours of the transfer window.
And Djilobodji may represent little more than a fall-back option as his pedigree is not quite at the standard Mourinho would wish for.
Meanwhile, Fabio Borini has completed a permanent switch to Sunderland after the Black Cats’ dogged pursuit finally paid dividends.
The 24-year-old Italian, who spent the 2013-14 season on loan at the Stadium of Light, severed his links with Liverpool when he signed a four-year contract for a fee understood to be £8m (€11m) with a further £2m (€2.7m) in add-ons.
Sunderland boss Dick Advocaat said: “Fabio is a great addition to the squad who will make our front line stronger. He can play in a number of attacking positions which is a huge benefit to the team.”
News of Borini’s arrival came on the same day the Black Cats allowed Emanuele Giaccherini to join Serie A side Bologna on a season-long loan.
Liverpool, meanwhile, have announced the signing of Nigerian forward Taiwo Awoniyi, who will immediately head out on loan with FSV Frankfurt. The 18-year-old is to spend the rest of the season on loan with the second division German side.
Liverpool are understood to have paid around £400,000 (€547,000) for Awoniyi’s services.
He said: “This is the biggest club in the world and I’m privileged to be here.
“I’ll try to develop myself very well so that when I come back to Liverpool I’ll be able to add value to the team.”
In other transfer news, Norwich have secured the services of Dynamo Kiev striker Dieumerci Mbokani, Manchester City defender Jason Denayer has joined Galatasaray on a season-long loan, Southampton have signed goalkeeper Harry Lewis from Shrewsbury while Hull striker Nikica Jelavic is closing in on a move to West Ham,
In the big transfer of the day outside of the Premier League Wolfsburg confirmed the signing of Julian Draxler from Schalke for a fee believed to be in excess of €30m.
Wolfsburg also completed the signing of Brazilian defender Dante from Bundesliga rivals Bayern Munich.
Marseille have announced the double loan signing of Juventus defenders Mauricio Isla and Paolo De Ceglie.