The local authority had previously said they were not suitable to adopt foster children, that were already in their care, because of their "concerning" views about same-sex parenting.
The couple and the council have not been named to protect the identity of the young people involved.
Christian Legal Centre, which has been working with the couple, told Premier the local council had accepted its initial rejection was wrong and agreed to reconsider its assessment.
The couple will now be assessed for suitability by someone outside the council.
The controversy started after a social worker told them that a same-sex couple were being considered as adoptive parents for the children.
The pair expressed concern saying the child's best interests are served by having "a mum and a dad".
They then reiterated their offer to adopt the children themselves, arguing that this would be the "best option for them and their emotional well-being".
Andrea Williams, Chief Executive of the Christian Legal Centre, said: "We are encouraged by the progress made this week, and grateful that the council has recognised its mistake and adopted a more appropriate stance. It is a step in the right direction.
"These developments highlight the value of Christians standing together, and where appropriate exposing the challenges that people are facing.
"There is still a long way to go, and we will continue to stand with this couple. Please pray with us that these young children are able to stay with them."