McKinnon wants to be tried in his home country because his jail sentence if convicted would be a lot less severe than in the US, where he is likely to face life imprisonment.
The US government accuses McKinnon of accessing and damaging 92 computers holding information on national defence, security and military operations. McKinnon has always claimed that he was looking for information on UFOs.
McKinnon attended a judicial review at the High Court this week to argue that his extradition will worsen his mental health, and that his diagnosis of Asperger's Syndrome in August last year had not been taken into consideration.
He has repeatedly discussed the strain he has felt since the UK's National High-Tech Crime Unit arrested him in 2002.
In August 2008, McKinnon asked the European Court of Human Rights to delay his extradition while he made a full appeal to the High Court. The application was denied and Jacqui Smith, UK home secretary at the time, ordered his extradition to go ahead. McKinnon won the chance to appeal against Smith's decision in January.
"The judgement of the judicial review on the Home Office decision to ignore the Asperger's syndrome aspects of the case, which finished yesterday, is 'reserved', so it might be published any time in the next three months, or perhaps even later," said a spokesman for McKinnon.
"His lawyers are seeking another judicial review about the Director of Public Prosecutions' decision not to prosecute in the UK, but it is unclear when, or even if, this will actually happen."
McKinnon's mother, Janis Sharp, discussed more details of the judicial review on Twitter, claiming that an intervention from the National Autistic Society had been "disallowed".
"UFOs talked about in court. Gary also believed the US allowed 9/11 to happen. This was discussed but not held against him," she wrote.
Sharp also said that the review had compared McKinnon's circumstances to that of Asperger's sufferer Bill Cottrell, who was convicted in the US in 2005 of conspiracy to commit arson and sentenced to eight years in prison.