The pair were sentenced to an indeterminate period in custody, with a minimum of five years, following the shocking incident in 2009 which became known as the Edlington Case, after the former South Yorkshire pit village where it happened.
A court order made at the time granted them anonymity until they are both 18.
As the younger of the two brothers approaches his 18th birthday, lawyers acting for the pair are seeking an injunction to extend their anonymity indefinitely, claiming that identifying them would breach various sections of the Human Rights Act.
The application will be heard at the High Court in London on Thursday.
The brothers' attack on the two boys, who were nine and 11, caused a wave of revulsion across the country and drew comparisons with the murder two-year-old James Bulger in 1993.
They lured their victims to a secluded spot in Edlington, near Doncaster, and subjected them to 90 minutes of violence and sexual humiliation.
The victims were throttled, hit with bricks, made to eat nettles, stripped and forced to sexually abuse each other.
Parts of the attack were recorded on a mobile phone.
Sentencing the brothers at Sheffield Crown Court, a judge told them their "truly exceptional" crimes amounted to torture,
Mr Justice Keith said: "The fact is this was prolonged, sadistic violence for no reason other than that you got a real kick out of hurting and humiliating them.
"The fact that you couldn't care less what happened to (the two boys) was a strong indicator you harm people simply because you want to."
The brothers admitted causing their victims grievous bodily harm with intent and other offences.
Referring to the parents of the victims, the judge said: "I have no doubt that they would have preferred to see (the two brothers) locked up for very much longer and I know that nothing can compare to the trauma the boys went through but I hope they will appreciate that five years is the very least (the brothers) will serve. They may well be in detention for much longer than that."
The brothers have now both been released and given new names, according to the Daily Mail.
The judge heard that the boys watched ultra-violent films as part of a "toxic home life" of "routine aggression, violence and chaos".
One of them watched the gruesome Saw movies when he was as young as 10 and was also familiar with the Chucky films as well as pornography DVDs. He also smoked cannabis from the age of nine and drank cider.
When they were sentenced, the victims' parents and the Sheffield Star newspaper asked the judge to allow them to be identified.
This move was opposed by the defendants, the secure units in which they were being held, the police and Doncaster Metropolitan Borough Council, and the judge refused to lift the reporting restriction.
He said: "The case has been regarded as raising important issues about the way children from dysfunctional families can go off the rails and about the lack of intervention at critical stages by the local authority social services department and other child protection agencies.
"The case has even been referred to at Prime Minister's Questions."
But he said that allowing the defendants to be identified could lead to problems for the brothers and their secure units, including them being "ostracised or harmed" by other inmates; problems for the brothers' family, including the cost of having to rehouse them if their identities were made known; and it could have have an adverse effect on their rehabilitation.
Social services in Doncaster, who were responsible for the brothers, came under severe criticism following the Edlington case.
Speaking immediately after the sentencing of the pair, the then interim director of the town's social services, Nick Jarman, offered "an unqualified apology on behalf of Doncaster Council for the admitted failings which led to this terrible incident".
He said he had arrived at the department and found "an organisation which was totally broken".
When the full serious case review (SCR) was published in 2012 it was heavily criticised by the then education secretary, Michael Gove, who said: "It is an example of how the current model of SCRs is failing."