Two judges in London will also consider a challenge by a second terror suspect, Khaled Al-Fawwaz, next Tuesday.
The men are seeking injunctions preventing their removal from the UK.
Pending the hearing of their applications by Sir John Thomas, President of the Queen’s Bench Division, and Mr Justice Ouseley, interim injunctions have been issued preventing their removal.
The latest legal action comes after Europe’s human rights judges this week rejected a bid for an appeal by Hamza and four other terror suspects, 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 Babar Ahmad, who was also subject to this week’s ruling, 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 Hamza and Ahmad, with Seyla Talha Ahsan, Adel Abdul Bary and Al-Fawwaz, 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.
A Home Office spokeswoman said: “The European Court of Human Rights ruled there was no bar to the extradition of these men.
“We will continue working to ensure they are handed over to the US authorities as soon as possible.”
Update 2: Lord Judge, the Lord Chief Justice, said it was “unacceptable” that any case should take eight years and still without a conclusion.
Speaking at his annual press briefing, Lord Judge said he could not comment on individual cases, but added: “Any case that takes eight years through a whole series of judicial processes to come to a conclusion – and you’ve made the point that it hasn’t yet come to a conclusion – is a source of real fury to me.
“We really can’t have cases taking that long to reach a conclusion. It’s not fair to anybody. It’s not right. And that’s all I can really say about it.”
It is not known on what grounds Hamza’s lawyers are arguing for the stay on extradition although Whitehall sources have previously suggested he ay try and argue he is not fit to travel to America.
A second terror suspect, Khaled al-Fawwaz, has also been granted a temporary injunction against his removal.
Lord Judge said: “The problem isn’t that there isn’t a limit on the number of appeals.
“The problem is that new points keep coming up and new points are taken and then they go through the process.
“What you need is a process in which all the points which need to be addressed are addressed once and then there is an appeal process and it comes to an end.
“Any case that takes eight years – unless there’s some extraordinary explanation, like the parties don’t want it, they’re not in a hurry – is unacceptable.
“People have to live their lives and they have to live their lives knowing where they stand.
“In the case of those sort of cases, alleged terrorist cases, well, actually, there’s a great public interest in disposing of them, fairly, justly, but with speed.”
Hamza, al-Fawwaz and three other terror suspects, who all face terror charges in America, lost their final appeal at the European Court of Human Rights on Monday.
The Home Office insisted it still aims to remove the pair “as soon as possible”.