Launching a legal challenge at London's High Court, a Reprieve lawyer said the decision "beggars belief".
The man, who Reprieve said is thought to be Yunus Rahmatullah, was one of two alleged terrorists captured by British soldiers in Iraq in February 2004.
The Ministry of Defence (MoD) said the Data Protection Act prevents it from disclosing their names.
This morning lawyers acting for Reprieve and Mr Rahmatullah's family lodged papers at the High Court in London to start a legal action to obtain disclosure.
Jamie Beagent, solicitor with Leigh Day & Co, said: "We have been compelled to issue legal proceedings because, despite repeated requests, the Secretary of State has refused to disclose even the identities of the two men.
"It frankly beggars belief that the Secretary of State has sought to rely upon the Data Protection Act to support his refusal."
Mr Beagent said it was hoped the application for judicial review would lead to Mr Rahmatullah's family "very soon" being able to contact and help him get access to the legal advice and representation he was currently being denied at Bagram.
The men were handed to the US authorities, who flew them to Afghanistan where it is thought they are being held at Bagram Air Base.
Former shadow home secretary and civil liberties campaigner David Davis said he believed the identities were being kept secret to avoid political embarrassment.
"If they are bad people, tell us who they are. I think the reason we are not being told is because it's politically embarrassing," he told BBC Radio 4's Today programme.
"These two people were captured by British troops, and released, I believe, accidentally to the Americans. They deserve a trial. We deserve to know what the truth is."
He added: "It's illegal to kidnap people and move them basically into permanent imprisonment without any recourse to law."
Reprieve last year named one as Pakistani national Amanatullah Ali, a rice merchant from the Shia sect of Islam, who was on pilgrimage in Iraq when he was captured.
Reprieve said Mr Rahmatullah, who is known as Saleh, is being held in the mental health wing of Bagram and has been unable to contact his family or a lawyer and was in a "legal black hole".
The charity said the British legal aid system will not allow his family to bring a case because there is not sufficient proof that Saleh was the prisoner rendered to Afghanistan, although proof is only available from the British Government.
Reprieve said the Government has threatened it will seek to impose costs if Reprieve brings legal action, a move which the charity says is designed to "intimidate".
Reprieve director Clive Stafford Smith wrote to Justice Secretary Jack Straw on Monday seeking assurance that the Government would withdraw this "threat" and is waiting for a response.
Mr Stafford Smith said: "The Government may think that bully-boy tactics will intimidate us. In truth, they merely steel our resolve.
"Yunus Rahmatullah's mother cries herself to sleep at night because the Government refuses to do the decent thing, and confirm or deny whether her son is alive, and whether her son was rendered from Iraq six years ago."
In a statement issued today through Reprieve, Fatima Rahmatullah said: "Our family was shocked when we learned that the British Government might have been behind Yunus' disappearance.
"My plea to the British Government is simple: Tell me whether you picked my son Yunus up, and gave him to the Americans.
"It is my basic right as a mother to know whether my son is still alive. I cannot bear further suspense. You have the power to help me recover my disappeared child."
In a statement to the House of Commons in February last year, then Defence Secretary John Hutton said the men were members of an extremist group Lashkar e Taiba (LeT).
LeT, which was responsible for the Mumbai attacks, is a radical Sunni group.
Today, an MoD spokesman said: "As the then Secretary of State made clear to Parliament in February 2009, the two individuals in question were insurgents captured in Baghdad as they posed an imperative threat to security of the Iraqi people and our Armed Forces. Their capture was legitimate and justified.
"We are prevented from disclosing their identity and, indeed, the identity of all other detainees, under the Data Protection Act.
"We can confirm however that the ICRC (International Committee of the Red Cross) has access to the two individuals and they are held in a humane, safe and secure environment within US detention facilities in Afghanistan, which meets international standards.
"The suggestion that Reprieve has been 'intimidated' by a threat of costs is ridiculous. Reprieve were seeking an assurance that the MoD would not pursue them for costs if they lost, but were clear that no reciprocal assurance would be provided. The MoD has therefore declined to give them this assurance."
Mr Hutton told the Commons the US government claimed the men were moved to Afghanistan because of a lack of translators, but Reprieve said its investigation found Rahmatullah spoke Arabic and could have been interrogated in Iraq.
Mr Hutton's statement followed years of denials that Britain had been involved in rendition.
In December last year, Mr Stafford Smith claimed ministers had made "factually incorrect" statements to MPs when they admitted Britain's complicity in their rendition.
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