Lawyers for the Islamist cleric say he is suffering sleep deprivation and the effects of being confined in an "unrelentingly harsh environment" in HMP Belmarsh.
In a last-ditch appeal after European judges said he could be extradited, Hamza's legal team say the move should be put on hold until he can undergo an MRI scan.
They say the Home Secretary has ignored their requests and instead " denounced" them for "abusing the process of the courts".
If the medical tests go ahead and doctors then agree Hamza is unfit to plead, his lawyers say it would be "oppressive" and a breach of his human rights to send him for trial abroad.
The application for an injunction was made on Tuesday just days after the most senior judge in the country, The Lord Chief Justice, said cases such as Hamza's that have dragged on for eight years.
In papers before two judges in London, 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."
Hamza is wanted in the United States with four others where they face terrorism charges.
Hamza, who lost both hands and an eye fighting the Soviet Union in Afghanistan, is charged in the US with 11 counts relating to the taking of 16 hostages in Yemen in 1998, advocating jihad in Afghanistan in 2001 and conspiring to establish a jihad training camp in Bly, Oregon in 2000-2001.
Last week a British judge granted a temporary ban against the hate cleric's removal after he lodged an application for an injunction.
It came after the five lost their final appeal at the European Court of Human Rights. They argued their detention in the US would amount to a breach of their human rights.