The court, based in Strasbourg, France, said in a release Tuesday the human rights of Hamza, Babar Ahmad, Syed Tahla Ahsan, Adel Abdul Bary and Khaled al-Fawwaz -- all jailed in Britain -- would not be violated they were sent to the United States.
The judges said the needed more information about the mental health of Haroon Aswat, an aide to Hamza, before ruling in his case.
In its opinion, the court said prohibitions against inhuman and degrading treatment, as set out in the European Convention on Human Rights, would not be violated if the five men were housed at a supermax prison in Florence, Colo.
"[Not] all inmates convicted of international terrorism were housed at ADX [Administrative Maximum Facility] and, even if they were, sufficient procedural safeguards were in place, such as holding a hearing before deciding on such a transfer," the court said in a release. "Furthermore, if the transfer process had been unsatisfactory, there was the possibility of bringing a claim to both the Federal Bureau of Prisons' administrative remedy program and the U.S. federal courts."
The court said the men should not be extradited from Britain to the United States until the judgment is finalized. The court said the men may appeal the ruling to the court's Grand Chamber within three months.
All six men were indicted on various terrorism charges in the United States between 1999 and 2006, the court said, including accusations of providing support to terrorists; conspiracy to kill, kidnap, maim or injure persons or damage property in a foreign country; and criminal conduct charges in the taking of 16 hostages in Yemen.
Bary and al-Fawwaz were indicted, along with Osama bin Laden and 20 others, in the 1998 bombings of U.S. embassies in Nairobi.Kenya. and Dar es Salaam, Tanzania, in which more than 269 people died.
British Prime Minister David Cameron said he was "very pleased" by the decision, ITV reported.
"It is quite right to have a proper legal process although sometimes one can get frustrated with how long they take," Cameron said. "I think deportation and extradition arrangements work promptly and properly particularly when people are accused of very serious crimes."