The daily paper states that the Istanbul based club close to agreeing terms with the Chelsea defender and that Djilobodji is keen on a move to Turkey.
Djilobodji joined Chelsea from the French side Nantes but is yet to make a league appearance for the Blues.
Chelsea manager Jose Mourinho recently stated that the decision to sign the Senegal international was not his choice.
“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.”
Beşiktaş defeated bitter rivals Bursaspor at the Atatürk stadium in their last Turkish Super Lig game.
Beşiktaş are currently first in the league two points clear of second placed Fenerbahçe and five ahead of reigning champions Galatasaray.