Diaz's father, Luis Manuel Diaz, and his mother were last weekend both kidnapped by the National Liberation Army (ELN). His mother, Cilenis Marulanda, was soon after rescued by police but his father has continued to be held captive.
But now, a week on from his kidnapping, Diaz's father will now be freed "as soon as possible".
"We salute Colombia and the Caribbean," the group said in a statement released to El Colombiano. 'We are in solidarity with the families who experience pain and tragedies due to unemployment, the criminal violence of paramilitarism and the abandonment of the State. The corruption of political clans is the main threat to the stability of the region.
"The Northern War Front has commands with economic missions and one of them carries out a privation of freedom, which, after being reported and verified that it is the priest of Lucho Diaz, is guided by his release because he is a family member of the great sportsman that we all love as Colombians.
"From that moment on, the release process begins and we want to avoid any incident. Firm in the search for the paths of peace, with the necessary transformations that the country needs. From the Caribbean."
Diaz has not played for Liverpool since the kidnapping but is in line to feature against Luton on Sunday.
Reds manager Jurgen Klopp told reporters on Friday: "He was in training two days ago, he had a session on Friday (with the squad), and will be part of the team. We must wait - if he feels right he will be here and train with us.
"The session he had with us, you could see that when he is with the boys it is fine, it’s OK. But you could see as well that he didn’t sleep a lot. So we will wait: it’s all about him, whether he makes himself available or not, and I will not force anything."