The Black Eagles president Fikret Orman is reported to have held talks with Chelsea in London and negotiations are rumoured to have gone to plan.
The Istanbul giants want to loan Djilobodji and have a history of doing business with Chelsea recently following the Demba Ba deal.
Beşiktaş could however, face competition from Lazio who the Express report are also preparing a loan move for the 26-year-old.
Djilobodji joined Chelsea from the French side Nantes but has yet to make a league appearance for the Blues and head coach Jose Mourinho went as far to claim that signing the Senegal international was not his decision.
“It was not my choice. It was the choice of somebody I trust completely,” Mourinho was quoted as saying by the Independent.
“I don’t know every player. My job doesn’t allow me to travel, doesn’t allow me to spend hours and hours watching players.”
The Eagles have made strengthening the defence a priority over the winter transfer window having conceded the second highest number of goals in the top three.
Beşiktaş are currently second in the Turkish Super League just one point behind league leaders Fenerbahçe.