Lawyers acting for honeymoon murder suspect Shrien Dewani are launching a bid to take his case to the Supreme Court.
It is the latest move to stop the 33-year-old businessman being sent back to South Africa to face trial over his wife's death.
Last month, Chief Magistrate Howard Riddle ruled at Westminster Magistrates' Court that Dewani should be extradited and rejected his attempt to stay in the UK for further hospital treatment for mental health problems.
Judge Riddle had given the go-ahead to Dewani's extradition in 2011. He had to reconsider the position after two senior High Court judges allowed an appeal in March last year. They were told Dewani was suffering from depression and post-traumatic stress disorder.
Sir John Thomas, president of the Queen's Bench Division, and Mr Justice Ouseley said it would be "unjust and oppressive" to remove him until he recovered - but it was plainly in the interests of justice that he was extradited as soon as he was fit.
Dewani's lawyers lodged an application at the High Court for the two judges to now certify that their ruling raised "a point of law of general public importance" for consideration by the Supreme Court, the highest court in the land.
Dewani is suspected of ordering the killing of his new wife Anni, 28, who was shot as the couple travelled in a taxi on the outskirts of Cape Town in November 2010. Last month, Judge Riddle said Dewani was still not fit to plead or stand trial at present, but there was evidence that he would receive the care he needed in South Africa.
So far three men have been convicted over Mrs Dewani's death. Last year South African Xolile Mngeni was convicted of premeditated murder for shooting her. Prosecutors claimed that he was a hitman hired by Dewani to kill his wife, something that Dewani has consistently denied.
Taxi driver Zola Tongo was jailed for 18 years after he admitted his part in the killing, and another accomplice, Mziwamadoda Qwabe, also pleaded guilty to murder and was handed a 25-year prison sentence.
Dewani's lawyers have stressed at various hearings that he will be willing to defend himself at trial once he is fit to do so.