In the 17-second clip, which appeared on Twitter on Wednesday, Enrich is seen having sexual intercourse with a young woman, who is simultaneously engaging in oral sex with ex-Aston Villa defender Luna.
In a statement posted to Enrich's Twitter account, the pair said: "In light of the circulation on social media of a sexually explicit video in which we, the undersigned, appear, we want to state that the video shows a recording of a private act which took place between fully consenting adults, within the scope of the freedoms we all enjoy.
"We deeply regret that an indiscretion for which we are not responsible has led to these images being published without our knowledge or, far less, our consent.
"We likewise regret the potential damage [not only] to our image but in particular to that of our club, whose colours we represent, and that it could have offended our supporters and the city of Eibar in general.
"We are aware that professional footballers should set an example in every walk of life, especially to children; this being the reason we apologise if this incident has caused harm to anyone.
"Equally, we want to apologise for the damage the circulation of this video could cause to the third person involved in the video.
"We wish to stress that, as Eibar players, an exemplary club that represents a set of fans and a city equally exemplary, we stand by the values that have been shown to us since we arrived in Eibar. We hope this incident does not in any way harm the image of the club, nor that of our team-mates."
Following the incident, a tweet posted on the official Eibar account in December 2015 has gone viral.
Minuto 69. Pase de la muerte de Antonio Luna a Sergi Enrich, que marca a portería vacía. 2-0 en Ipurua. AUPA EIBAR!!!— SD Eibar (@SDEibar) December 16, 2015
Translated into English, it reads: "Minute 69. Killer pass from Antonio Luna to Sergi Enrich, who scores in the open goal. 2-0 in Ipurua. Aupa Eibar."