A second suspect, Khaled Al-Fawwaz, has also began a challenge.
Pending the hearing of their applications at the High Court in London – at a date to be fixed urgently – a judge has issued interim injunctions preventing their removal.
A spokeswoman for the Judicial Office confirmed that both men were “seeking injunctions preventing their removal from the UK”, adding: “A High Court judge has considered the applications on the papers and adjourned the cases to a hearing in open court.
“The judge has issued interim injunctions preventing their removal prior to those hearings. The judge has directed the hearings be fixed urgently.”
The move came after Europe’s human rights judges this week rejected a bid for an appeal by Hamza and four others, paving the way for their extradition.
The 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”.