The Ghanaian midfielder is on loan at the Spanish champions from Chelsea and put in some impressive performances in the opening weeks of the campaign.
However, Essien has not played since November 11 and Real had previously kept quiet about the reasons for his absence until assistant coach Aitor Karanka explained the situation.
"He has had an allergic reaction to some medicine," Jose Mourinho's right-hand man told reporters.
Karanka also confirmed that Real are not currently putting a timescale on when Marcelo and Gonzalo Higuain will return from a broken foot and thigh injury respectively.
"Marcelo's injury is a long one and it depends on how things evolve," he added. "Today or tomorrow he will come out of the cast.
"Gonzalo's injury is muscular. We have to be careful and judge the best time for him to return."