The Foreign Office is asking the high court to sit in secret when former foreign secretary Jack Straw faces a damages claim over his alleged role in the abduction and torture of a Libyan dissident and his pregnant wife.

Having failed in an attempt to have the case against Straw struck out, government lawyers are now seeking to have the case heard behind closed doors under controversial new secret justice measures.

The Guardian understands that the current foreign secretary, Boris Johnson, has signed the application that has been made to the high court.

Straw is a defendant in the case alongside Sir Mark Allen, a former senior MI6 officer, MI6 itself, MI5, and the Foreign Office.

The case is being brought by Abdel Hakim Belhaj and his wife Fatima Bouchar, who were detained in Bangkok in March 2004 following a tip-off from MI6, and flown to Tripoli against their will in a so-called extraordinary rendition operation.

Documents discovered in Tripoli during the 2011 Libyan revolution include a fax from Allen to Muammar Gaddafi’s intelligence chief, Moussa Koussa, in which Allen took credit for sharing the intelligence that enabled the couple to be abducted.

Other papers appear to show that the operation was the culmination of more than two years of cooperation between MI6 and Gaddafi’s external security organisation.

Belhaj and Bouchar have since described how masked CIA officers detained them at Bangkok airport, strapped them to stretchers and placed them aboard an aircraft that flew them to Tripoli.

Bouchar, a Moroccan national, was imprisoned for five months, and released shortly before giving birth to a son.

Her husband, who was the leader of an anti-Gaddafi force, the Libyan Islamic Fighting Group (LIFG), was held for six years and says he was repeatedly tortured.

Belhaj was also interrogated by British intelligence officers. He says then when he silently indicated to these officers that he was being tortured, one, a woman, nodded, while her male companion gave a thumbs-up signal.

The couple are suing for damages for what they say was the defendants’ involvement in their abduction, false imprisonment and mistreatment, and misfeasance in public office.

Their lawyers argue that as Straw was the government minister responsible for MI6 at that time, he either authorised the rendition operation, or took no steps to prevent it.

They also say there is clear documentary evidence that Allen led MI6’s contacts with the Libyan government and was the senior MI6 officer with responsibility for the rendition operation.

Straw has said that he was opposed to extraordinary rendition, was not complicit in it and did not turn a blind eye to it. Allen has declined publicly to comment.

At a hearing next month, the defendants’ lawyers will argue that it should be heard in secret under the terms of the Justice and Security Act, which was passed after damning details of British government involvement in rendition and the mistreatment of detainees emerged in two previous civil cases.

The secrecy surrounding the case contrasts sharply with a claim currently being brought by two former detainees of the CIA in US courts, where the video-taped depositions of CIA officers and military psychologists have been made available to the media.

Asked why the Foreign Office wished to case to be heard in secret, a spokesperson for the department said: “As legal proceedings are ongoing it is not appropriate to comment.”

Cori Crider of the legal charity Reprieve, which representes the couple, said: “The government’s move smacks of desperation. The evidence showing that MI6 helped to abduct my clients to prop up Blair’s ‘deal in the desert’ has been all over the press for years – this horse is halfway down the meadow and still the government is trying to shut the stable door.

“After extensive investigation we know who abducted my clients; who in MI6 was most involved; the spies MI6 recruited to make it possible; we even have the fuel bill for the kidnap flight, for goodness’ sake. Given that we quite literally have the receipts in this case, what’s the legitimate basis for hiding this trial from the British public? There isn’t one. This is a naked effort to spare Jack Straw, Mark Allen, and MI6 from embarrassment for this disgusting episode: the brutal abduction of a Gaddafi opponent and his pregnant wife.”

Three weeks after the abduction of Belhaj and Bouchar, another leading member of the LIFG, Sami al-Saadi, was abducted in Hong Kong and flown to Tripoli along with his wife and four children, the youngest a girl aged six.

In between the two rendition operations, Tony Blair paid his first visit to Gaddafi, and declared that Libya had recognised “a common cause, with us, in the fight against al-Qaida extremism and terrorism”.

At the same time, in London, the Anglo-Dutch oil giant Shell announced that it had signed a £110m deal for gas exploration rights off the Libyan coast. Al-Saadi and his family also sued the Foreign Office, but settled their case in 2012 after the government agreed to pay £2.2m in damages.

Belhaj has offered to settle his claim for just £1, but insists that he and his wife must also receive an unreserved apology from the British government. Instead, government lawyers spent four years arguing that the case should be struck out. These attempts ended in January, when seven justices of the supreme court ruled unanimously that the claim must be heard by the courts.

A Freedom of Information Act request subsequently established that the government’s failed attempt had cost the taxpayer £1.6m in legal fees.

Separately, lawyers for the couple are seeking permission for a judicial review of the decision by the Crown Prosecution Service not to charge Allen with any criminal offence.

That decision is said to have angered some police officers at Scotland Yard, which believed it had built a powerful case during an investigation that lasted four years.