If judges refuse to extend injunctions preventing their removal then the Home Office is expected to move immediately and put them on US-bound flights before fresh legal objections can be lodged.
Sir John Thomas, president of the Queen's bench division, and Mr Justice Ouseley have promised to deliver their decision after considering three days of argument at the Royal Courts of Justice in London.
The cases involving Hamza, Babar Ahmad, Syed Talha Ahsan, Khaled al-Fawwaz and Adel Abdul Bary have together taken more than eight years and involved appeals up through the hierarchy of British and European courts.
Hamza, 54, who was jailed for seven years for soliciting to murder and inciting racial hatred, has been fighting extradition since 2004. His lawyers oppose deportation on the grounds that he is suffering from memory-loss and depression and is unfit to plead. They are seeking permission for the former iman at Finsbury Park mosque in north London to be given an MRI scan to assess his medical condition.
Ahmad, 37, a computer expert, and Ahsan, 33, are accused of raising funds for terrorism through a website. Lawyers for the two men are challenging the director of public prosecution's decision not to charge them with offences allegedly committed in the UK.
Al-Fawwaz, alleged to have been an aide to Osama bin Laden in the 1990s, is seeking disclosure of an 800-page MI6 document relating to the debriefing of another suspect, which, his lawyers maintain, would undermine the charges against him.
Adel Abdul Bary, 52, is also said to have worked closely with Bin Laden. In court, his barrister argued that conditions in US high security jails would breach his rights under the European convention of human rights – a claim already dismissed by the Strasbourg court.
Writing for the Guardian's Comment is Free website, Ahmad said this week: "I expect to be extradited to the US imminently. Since my detention without charge began in August 2004, millions of pounds of British taxpayers' money has been spent on keeping me in a high-security prison all these years, on legal costs and parliamentary business.
"... As a British citizen who has lived since birth in Britain, studied, worked full-time and paid taxes, if I am accused of any offence here in Britain I expect at the very least to face trial here in Britain."
James Eadie QC, representing the government, has dismissed legal arguments aimed at delaying extradition as "much too late" and an "abuse of process". He agreed that if the court rejected the men's applications then the home secretary, Theresa May, "would be entitled to move instantly".
Supporters of the men are planning demonstrations against their removal.
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