Radical cleric Abu Hamza and other terror suspects will attempt to persuade the High Court to halt their extradition from the UK to the US.
Two judges in London will consider challenges by Hamza, Babar Ahmad, Khaled Al-Fawwaz and Adel Abdul Bary, it was confirmed by the Judicial Office.
The men - who will not be attending the proceedings - are seeking injunctions from the court preventing their removal. 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. Computer expert Ahmad has been held in a UK prison without trial for eight years after being accused of raising funds for terrorism.
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.
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.
Ahmad and Ahsan are accused of offences including providing support to terrorists and conspiracy to kill, kidnap, maim or injure persons or damage property in a foreign country.
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.