Lawyers acting for two men who were jailed for 10 years without trial and say they were tortured after being seized by British troops in Iraq, plan to take the body responsible for investigating the case to court, claiming it is failing to seek potentially vital evidence from the US.

Yunus Rahmatullah and Amanatullah Ali, both Pakistanis, were captured in Iraq in 2004 by British special forces and handed over to American soldiers. Their lawyers have asked the Iraq Historic Allegations Team (IHAT), set up by the Ministry of Defence in the wake of mounting claims of ill-treatment of detainees, to demand information about the case from the US, notably the CIA.

They want IHAT to ask for an uncensored copy of the report by the US Senate intelligence committee into CIA torture, the Guardian has learned. A redacted 480-page summary of the report was published at the end of last year but omitted any reference to US operations involving America’s allies, including the UK.

The British government says it was consulted about the redactions but said they were made on grounds of “national security”, a phrase Whitehall interprets extremely broadly.

It is understood the two Pakistanis were detained and interrogated by the CIA. “It follows that there is likely to be evidence about Mr Rahmatullah and Mr Ali in the material sought”, the men’s lawyers, Leigh Day, have told IHAT.

They say IHAT is under an “investigative duty” to ask for the complete US Senate committee’s report on torture and that it would seek a court hearing if it fails to do so.

IHAT told the Guardian that it “may ask for a copy of the report if this becomes relevant to an IHAT line of inquiry”.

However, it added that it would be “inappropiate” to share information with the lawyer for the two men. “We would not be prepared to hand over the relevant correspondence to prosecution witnesses or their lawyers during a live investigation as there is a real risk that this could taint their evidence, putting any future criminal trial at risk,” IHAT added.

Kat Craig, legal director at Reprieve, the law charity supporting the two men, told the Guardian on Thursday: “Yunus Rahmatullah and Amanatullah Ali suffered a horrific, 10-year ordeal of torture and detention at the hands of the UK and US. These men deserve answers and justice – and the CIA torture report is a crucial piece of evidence in that effort. IHAT’s refusal to request a copy of that report is therefore inexplicable. Who should the victims of torture and rendition look to for accountability, if they are let down like this at every turn?”

Rahmatullah and Ali were was seized in Iraq in 2004 in an incident that was kept secret from ministers and only disclosed to MPs five years later. They were released by the US without charge last year. They are seeking to sue the Ministry of Defence and the Foreign Office, accusing them of responsibility for their torture and abuse.

The MoD and FO argue that if the high court allows the case to go ahead, the UK’s defence and security relationship with the US would be seriously harmed. They argue that the “act of state” doctrine means that a British court cannot question the activities of US troops.

The two Pakistanis, one Sunni, the other Shia, are believed to have been held first at Camp Nama, a secret detention facility at Baghdad airport that British troops helped to run. They were later transferred to Iraq’s notorious Abu Ghraib jail before being rendered to the Bagram “black prison” in Afghanistan.

In 2013 the UK supreme court described Rahmatullah’s treatment by UK and US forces as unlawful and a possible war crime. “The, presumably forcible, transfer of Mr Rahmatullah from Iraq to Afghanistan is, at least prima facie, a breach of article 49 [of the fourth Geneva convention],” it said. But the court agreed with the UK government that it was unable to enforce his release.

John Hutton, the defence secretary at the time, admitted to the Commons in 2009 that British officials knew about the transfer of the prisoners in 2004, although the government had previously denied having any knowledge of the case. He said they were members of Lashkar-e-Taiba, (LeT) a banned organisation that he said was linked to al-Qaida.