Following a ruling by the European High Court that would allow his extradition to the United States on terrorism charges, Abu Hamza al-Masri is seeking relief from the European Court of Human Rights, saying that in sending him to the US, he would face inhumane treatment.
Legal sources have told the BBC that he had new evidence to support his case against extradition.
It is said Mr Al-Fawwaz has been delisted as a terrorist by the US authorities and the home secretary has been invited to reconsider her decision to approve his extradition.
It is understood that a judge has already reviewed the papers lodged with the court.
BBC home affairs correspondent Danny Shaw said lawyers for Abu Hamza and Mr Al-Fawwaz would be told within hours whether they would be granted fresh hearings.
A Home Office spokesman 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're handed over to the US authorities as soon as possible".
The European court's decision on Monday meant that Abu Hamza and the four others were facing trials in the US after delays going back to the late 1990s.
It was expected the extradition of the men was likely to happen within weeks.
In April, the Strasbourg court ruled there would be "no violation" of the suspects' rights if put on trial in the US. In Monday's ruling, the court refused men permission to launch an appeal against that decision.
Abu Hamza is wanted over allegations he plotted to set up a terrorist training camp in the US and was involved in kidnapping Western hostages in Yemen.
This case has been allowed to drag on far too long by British authorities. Time to send him to the US and let us deal with him. It's obvious the British government wants to wash its hands of Mr. Hamza.