The court will rule on whether six men, including Abu Hamza, should be extradited on terrorism charges to the US.
John Bolton, the American ambassador to the UN under George Bush, said: "Britain should renounce the jurisdiction of this court. It's a question of what do British people want to do? Do you want to be an independent nation, or do you want to be a county in Europe?
"This is just another example of Britain's mistake in allowing European institutions to develop to the extent they have. It is yet another infringement on British sovereignty that undercuts its ability to cooperate with the United States.
"It also calls into question the ability of Europe as a whole to be an effective partner in the war against terrorism."
Estimates obtained by The Daily Telegraph suggest that it has cost the taxpayer £2.6 million to keep the six men in high-security jails. A further £1.5 million is estimated to have been spent on legal costs, including legal aid for the men, and thousands more on benefit claims for their families.
The men have been fighting extradition on human rights grounds for up to 14 years.
Hamza is wanted in the US on 11 charges related to taking 16 hostages in Yemen in 1998, promoting violent jihad in Afghanistan in 2001 and nationals wanted in the US - Haroon Rashid Aswat, Babar Ahmad and Seyla Talha Ahsan - are also fighting extradition, alongside Egyptian-born Adel Abdel Bary and Khaled al-Fawwaz, from Saudi Arabia.
Al-Fawwaz, allegedly a close associate of Osama bin Laden, and Bary are both wanted in connection with the 1998 embassy bombings in East Africa which killed 223 people.
All six have been indicted on charges of alleged terrorism in the US and argue that the conditions of their detention in a "supermax" prison would amount to inhuman and degrading treatment under Article 3 of the Human Rights code.
The regime, originally designed to remove troublesome prisoners from the main prison population, involves life in solitary confinement in a cell measuring 11ft 5ins by 6ft 6ins for 23 hours a day. Prisoners are fed through the door and allowed out for only one hour a day to exercise alone in a concrete pen.
Hamza, who would not be incarcerated in one of the prisons because of his disability, says that a sentence of life without parole would amount to a breach of his human rights.
Last night Conservative MPs urged the Government had to replace European human rights laws with a British Bill of Rights.
Dover MP Charlie Elphicke said: "This is a crazy waste of money, and shows once again why we need a British Bill of Rights. People will find it incomprehensible."
Nick de Bois MP, a member of the Commons justice committee, added: "We have a very distorted system that allows taxpayers to pay an obscene amount of money as a result of the discredited European justice system exploited by these suspect terrorists."
Mark Pritchard MP, said Hamza's original student visa must be "the most costly ever granted", adding: "If he had been asked to return to Egypt after his studies in Brighton in the 1980s, the British taxpayer would not now be footing this huge bill."
Keith Vaz MP, the Labour chairman of the Commons' Home affairs select committee, said it was a "cause for concern" that it was taking so long to extradite people Britons to face charges, compared with France.
He said: "The contrast between what President Sarkozy has been doing - using his powers to deport people - where we take a long time is a cause for concern.
"We need to look at primary legislation. We need to examine what is happening in other countries. We are the only country that can remove people slowly."
Charles Stimson, Deputy Assistant Defence Secretary for Detainee Affairs under George W. Bush, added: "We have a special relationship and it needs to continue. But I'm hopeful that the momentum is toward bringing them here.
"This is a special case and Abu Hamza is clearly dangerous and we have legitimate law-enforcement interests in bringing him here to face federal courts."
Richard Perle, another former Bush administration official, said: "I sincerely hope the court does not make this judgment, as I am sure it would then be invoked in every other case. But if it forced to choose, then the UK should certainly ignore it. I don't know of any mechanism by which the court could enforce it."
He added: "The relationship between the US and UK is large enough to survive this, but it would do real damage."