If the appeal goes ahead, then Infosecurity understands an extradition of McKinnon would be unlikely to happen before next spring.
McKinnon, who has been diagnosed with Aspergers syndrome, has admitted his hacking exploits, but has claimed all along that he carried out his unauthorised accesses merely to prove the existence of UFOs and the fact that they have been investigated by the US military.
The Supreme Court application for an appeal comes as a cross party group of MPs attempted earlier this month to apply fresh political pressure on the current Home Secretary Alan Johnson to halt the extradition of McKinnon.
The delegation of three Labour, Conservative and Liberal Democrat members told Johnson in a Home Office meeting last week that he has the "power and the duty" to step in to prevent McKinnon from being sent for trial in the USA.
David Davis, the former Shadow Home Secretary, the Liberal Democrat home affairs spokesman Chris Huhne and Michael Meacher, the former Labour minister, all joined forces for the meeting looking at the case of McKinnon.
According to newspaper reports, they presented Johnson with a detailed legal opinion challenging the government's earlier claim that it has no power to intervene in the extradition which has already been agreed by the courts.
As reported previously by Infosecurity, in July of this year, McKinnon lost an eleventh hour High Court challenge to the move to extradite him to the USA, claiming that it would worsen his medical condition and thus represent inhuman treatment breaching his human rights.
Under the Supreme Court appeal application, the McKinnon legal team claim the court should hear an appeal under the European convention on human rights.
The legal team are arguing that, because the court has agreed to hear an appeal against extradition to the USA brought by Ian Norris, a businessman who has prostate cancer, that ruling should apply to McKinnon too.
Liberty, the civil rights group, has expressed its support for McKinnon's case, and has said that there may be "compelling personal reasons why a defendant should not be sent abroad for trial".
McKinnon is facing extradition to the USA on charges of perpetrating what is claimed by one US prosecutor to be the "biggest military computer hack of all time."
Following legal hearings in the UK it was decided in July of 2006 that McKinnon should be extradited to the USA.
In February 2007 his lawyers argued against the ruling in an appeal to the High Court in London, which was turned down in April of that year.
In July 2007, the House of Lords agreed to hear the appeal and in June 2008 the Law Lords began hearing the McKinnon case.
The judgement, which was delivered a month later, ruled that McKinnon could be extradited to the USA. A subsequent appeal to the Law Lords was rejected.
His legal team then lodged a further appeal, which was granted, on the basis that McKinnon had been diagnosed in August 2007 as having Aspergers syndrome, a form of autism.