It is thought this is the first step to the child identifying as a gender other to the sex they were born.
Their family lawyer has expressed concerns the child could be taken into care if the family do not comply with the social worker's guidance.
Speaking to The Sunday Times anonymously, the child's mother said: "The rights of parents in the UK are being eroded, especially those who have traditional Christian values."
The mother said she felt the child was not old enough to make a choice about their gender.
Social workers say the parents Christian beliefs in traditional male and female roles means they do not accept their child may be transgender.
The Christian Legal Centre, supporting the family, has reported that Social Services and Child and Adolescent Mental Health Services told the parents that unless they allowed the name change, their child would be at risk of suicide.
Christian Legal Centre Chief Executive, Andrea Williams, said: "The new transgender cultural movement is tantamount to The Emperor's New Clothes.
"In gender, there is an objective reality but forces are at work in seeking to undermine the obvious.
"The authorities are forcing an agenda that is untrue and harmful to our children.
"This case demonstrates a shocking disregard for parental authority; no one is listening to what the parents want or have to say; they know the child best; they have the child's best interests at heart."
The mother has said the child may make a choice about their own gender when they are an adult at 18-years-old.
Child and Adolescent Mental Health Services did not comment.
