Mr McKinnon, from north London, admits hacking US military computers but says he was looking for evidence of UFOs. If he is convicted in the US, he could face up to 60 years in jail.
The Home Office said the matter would be discussed at a hearing at the High Court on 24 July. At the last court hearing on July 5, judges were told Theresa May was “close” to making a decision.
But Mrs May said she was “personally concerned” he had not been examined by a Home Office-appointed medical assessor, to decide whether there was a risk of suicide if extradited.
His family say the Home Office expert, Professor Thomas Fahy, has no experience in uncovering suicidal tendencies in Asperger’s syndrome patients.
Ms Sharp told BBC Three Counties Radio: “It is not a refusal, he had no choice – it is an impossibility because the assessment they want him to have is by someone who has no experience and wouldn’t be able to diagnose his suicide risk.”
Mr McKinnon, 46, had three medical examinations in April by three leading experts in Aspergers and suicidal risk, Professor Simon Baron Cohen, Professor Jeremy Turk and Dr Jan Vermeulen.
They concluded Mr McKinnon was at extreme risk of suicide if extradited and he was currently unfit for trial.