The report claims that the Black Eagles have already started holding negotiations with the Blues and that Djilobodji is keen on a move after failing to settle in.
The Senegal international has made just one appearance all season and that was in the Capital One Cup against Walsall.
Djilobodji has yet to make a single league appearance since joining the Blues from French side Nantes.
Former Chelsea boss Jose Mourinho revealed that he did not even want to sign the defender during his time as Blues boss.
“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 Black Eagles are currently in the market for a defender over the winter transfer window after conceding the second highest number of goals in the top six (18).
Beşiktaş are currently first in the league one point clear of Istanbul rivals Fenerbahçe and nine ahead of reigning champions Galatasaray who are currently in third place.