The two family members are represented by solicitor-advocate Robin Makin. Sir James Munby, President of the High Court's Family Division, is dealing with preliminary issues in the application to vary or discharge an injunction the High Court granted in 2001.
Venables has been living anonymously since his release from a life sentence for the kidnap, torture and murder of 2-year-old James 25 years ago.
James was murdered by 10-year-olds Venables and Robert Thompson after they snatched him from a shopping centre in Bootle, Merseyside, in February 1993.
Venables and Thompson were later granted lifelong anonymity by a High Court judge.
After release they have lived under new identities. But Venables has since been convicted and sent back to jail over indecent images of children.
In February, he was jailed for three years and four months after admitting surfing the dark web for extreme child abuse images and possessing a "sickening" paedophile manual.
He was charged after police found more than 1000 indecent images on his computer. It was the second time he had been caught with such images and when he was arrested he told police he was plagued by "stupid urges".
Fergus and Ralph Bulger attended the Old Bailey when Venables was sentenced in February.