The father and uncle of the murdered toddler James Bulger have lost a legal challenge to overturn the lifelong anonymity protecting one of his killers, Jon Venables.

Ralph and Jimmy Bulger argued that information about Venables that they claimed was “common knowledge” should be made public. But on Monday the president of the high court’s family division, Sir Andrew McFarlane, ruled that the injunction granting anonymity should be maintained.

Describing it as a “wholly exceptional case”, McFarlane said: “There is a strong possibility, if not a probability, that if his identity were known he would be pursued resulting in grave and possibly fatal consequences … My decision is in no way a reflection on the applicants themselves, for whom there is a profoundest sympathy. The reality is that the case for varying the injunction has simply not been made.”

Venables and Robert Thompson were 10 when they carried out one of the UK’s most notorious child murders. They were convicted of killing two-year-old James in 1993. Before their release on parole, they were given new identities, protected by a court order.

Venables has since been sent back to prison on charges related to indecent images of children. In February last year, he was jailed for three years and four months after admitting surfing the dark web for extreme child abuse images and possessing a paedophile manual.

Challenging the 2001 injunction, solicitor-advocate Robin Makin, for the Bulgers, told the high court that something had “gone wrong” with Venables’ rehabilitation and that the claimants, as victims, should be able to scrutinise his handling by the authorities.

He stressed they did not want the order to be discharged altogether but wanted it to be varied so that some information could be revealed without the threat of prosecution.

This included names used by Venables prior to his return to custody in 2017, his whereabouts and activities up to that date, and details of any release from custody – and subsequent recall to custody – following his 2018 conviction.

Makin argued that such details would help inform public debate about rehabilitation of offenders and help protect the public from Venables.

The attorney general’s office had, along with Venables, opposed variation to the injunction, which it argued was both necessary and justified. McFarlane said he was not convinced that Venables’ identity was already public and even if it were that did not justify giving it “a higher profile than is currently the case”.

The judge said the injunction was designed to protect Venables from “being put to death”. However, he said he would permit “any agreed relaxation regarding the Ministry of Justice’s ability to disclose, from time to time if JV [Venables] is in custody or out on licence”.

McFarlane denied the Bulgers leave to appeal but, speaking after the hearing, their lawyer said they would consider applying directly to the court of appeal to challenge the decision.

Makin said: “The authorities seem to be hell-bent on protecting JV regardless of the risk to others, and this has been a primary driving force behind Ralph and Jimmy’s application.”

He accused the authorities of being out of touch with what information was already public and said that if they really wanted to protect Venables they should be taking action against social media companies such as Facebook, Google and Twitter, on whose platforms he said the convicted murderer had already been identified.

James’s mother, Denise Fergus, distanced herself from the application when it was lodged, saying she feared it could lead to vigilante action.