Hamza and four other radical Muslim clerics, have asked the court for a temporary injunction blocking his extradition to the United States, the BBC reported Tuesday.
Hamza's lawyers said his client's poor health could be caused by his detention in "an unrelentingly harsh environment," and that prompt medical attention was necessary.
Hamza and the other terror suspects jailed in Britain are wanted in the United States, where they are accused of planning a terror training camp in the United States and assisting hostage-taking in Yemen.
The court has on file applications for injunctions against the deportation of Abu Hamza, Babar Ahmed, Khaled al-Fawwaz and Adel Abdul Bary, The Guardian said. The Royal Courts of Justice list did not mention the fifth suspect, Syed Talha Ahsan.
To avoid extradition, Hamza and the others must demonstrate "new and compelling" reasons for them not to be sent, the BBC said.
The applications came after Keir Starmer, director of public prosecutions, said he will not support a private criminal action against two of the detainees, clearing a legal obstacle to their deportation, The Guardian reported.
Last week, the European Court of Human Rights rejected appeals to its upper chamber, agreeing with an earlier ruling that the human rights of the five suspects would not be violated by life sentences or solitary confinement in a so-called "supermax" prison in the United States.
However, the decision to extradite the men has meet resistance in Britain, The Guardian said.
"In the case of Babar Ahmad if there was a crime committed it was committed in this country," London Mayor Boris Johnson said in a statement. "There is absolutely no reason why this gentleman should not be produced before the British courts, arraigned and asked to answer to whatever his crimes are here in [Britain]. "
Hamza, jailed for seven years for soliciting to murder and inciting racial hatred, has been fighting extradition since 2004. Ahmed has been held in a British prison without trial for eight years, accused of fundraising for terrorism through a website. Fawwaz and Bary are accused of being aides to Osama bin Laden. Ahsan is being held for his alleged involvement with a series of terrorist websites.