Lawyers for Abu Hamza have argued that his poor health means he should not be extradited as he and four other terror suspects took their cases to the High Court in London in an attempt to halt their extradition to the US.
Two judges are considering challenges by Hamza, Babar Ahmad, Khaled Al-Fawwaz and Adel Abdul Bary.
The men - who will not be attending the proceedings - are seeking injunctions from the court preventing their removal to the US.
Abu Hamza and other terror suspects took their cases to the High Court on Tuesday
It is understood that an application by a fifth suspect, Syed Ahsan, will also be heard by Sir John Thomas, president of the Queen's Bench Division, and Mr Justice Ouseley.
The latest legal action comes after Europe's human rights judges recently rejected a bid for an appeal by Hamza and the others, paving the way for their extradition.
A panel of five judges threw out their request to the Grand Chamber of the European Court of Human Rights.
Hamza, who was jailed for seven years for soliciting to murder and inciting racial hatred, has been fighting extradition since 2004.
Abu Hamza's legal team has been arguing the radical cleric needs an MRI scan to establish his health.
In papers put before the two judges, Hamza's QC, Alun Jones, argues that there is "uncontradicted medical opinion that a scan is medically necessary".
Mr Jones adds: "If the applicant (Hamza) is unfit to plead, or arguably so, it will be argued that it would be oppressive to extradite him within the meaning of Section 91 of the 2003 Extradition Act."
The QC says a judge referred to Hamza's "very poor health" at an extradition hearing in 2008.
"Over four years later, it appears there has been, or may have been, a further deterioration, perhaps attributable to sleep deprivation and the continued confinement of the appellant in an unrelentingly harsh environment.
"The responsibility of the (Home Secretary) is a relevant factor here," says Mr Jones.
Hamza has been charged with 11 counts of criminal conduct related to the taking of 16 hostages in Yemen in 1998, advocating violent jihad in Afghanistan in 2001, and conspiring to establish a jihad training camp in Bly, Oregon, between June 2000 and December 2001.
Bary and Al-Fawwaz were indicted - with Osama bin Laden and 20 others - for their alleged involvement in, or support for, the bombing of US embassies in Nairobi and Dar es Salaam in 1998. Al-Fawwaz faces more than 269 counts of murder.
Khaled al-Fawwaz's legal team is arguing that new evidence casts doubt on his need for extradition.
The judge has said that the case is unlikely to conclude on Tuesday, reported Danny Shaw for the BBC.
The campaign to try Ahmad in the UK has had much public support
Computer expert Babar Ahmad has been held in a UK prison without trial for eight years after being accused of raising funds for terrorism.
On Monday, a private prosecution of Babar Ahmad and Syed Ahsan was blocked by the director of public prosecutions because "no real evidence has been provided and any case against the men would be 'bound to fail' in a British court."
Businessman Kevin Watkin made the attempt to bring legal proceedings against the pair in the UK to avoid "outsourcing the country's criminal justice system" to the US.
The campaign to try Ahmad in the UK has had much public support, with Mr Watkin telling Channel 4 he had spent £250,000 of his own money fighting the extradition treaty, with he and other campaginers having budgeted £150,000 for the Ahmad case and spent £20,000 so far.
"British people should be subject to British law. I am an international businessman and I always stand up for the rights of people who are getting abused," he told the news channel.
The pair are accused of being involved in a website which encouraged terrorism and which, while operated from London, was hosted in the US.
After the ruling in Europe, the Home Office said the five men would be "handed over to the US authorities as quickly as possible".
Between 1999 and 2006, the men were indicted on various terrorism charges in America.