Mr McKinnon, who faces trial in the US over hacking into military computers, will find out today if Mrs May will offer him a last-minute reprieve.
The Asperger's sufferer has been fighting against extradition for more than a decade and medical experts have warned he will kill himself if he is sent to the US to face trial.
Prime Minister David Cameron and his deputy Nick Clegg have previously condemned plans to send Mr McKinnon to the US.
His legal team, led by lawyer Karen Todner, hopes Mrs May will block extradition amid predictions he could be jailed for 60 years in America.
Ms Todner said: "It has been a long 11-year battle to fight this extradition and we wait with anxiety, but hope, that the Home Secretary will uphold the promises previously made by Mr Cameron and Mr Clegg.
"We hope our elected Government will uphold the promises made while in opposition and will prevent Mr McKinnon's extradition."
She said medical evidence shows the 46-year-old is suffering from a serious mental disorder and there is a serious risk of suicide if extradited.
Mr McKinnon's mother, Janis Sharp, who has headed a campaign against her son's extradition, said: "I feel upset, I feel scared, I feel it's a complete waste of life.
"For 10 years Gary has lived a zombified life. It has destroyed him, it has destroyed our lives.
"My terror is that Gary wouldn't last five minutes away from home. Doctors have said he will take his own life, not could or might, if this extradition proceeds and there's nothing they can do to stop it.
"I've never said that Gary shouldn't be tried for what he has done, but try him in this country. When an American commits a crime and they are physically in America, they are tried in America. Should a British person have less rights? Absolutely not."
Mr McKinnon, who was born in Glasgow but now lives in Wood Green, north London, was arrested in 2002, and then again in 2005, before an order for his extradition was made in July 2006 under the 2003 Extradition Act.
That triggered three successive applications for judicial review and questions about the fairness of the UK-US extradition treaty, which critics claim is one-sided.
In a House of Commons debate in December, Tory MP Dominic Raab said: "Gary McKinnon should not be treated like some gangland mobster or al Qaeda mastermind."
Jago Russell, chief executive of Fair Trials International, said: "Gary McKinnon's case has highlighted the massive human impact of our extradition laws, which affect hundreds of people in Britain every year.
"Our current arrangements have some serious defects."
Shami Chakrabarti, director of the civil rights group Liberty, urged Mrs May to "show compassion and common sense and stop him being sent to the US".
She added: "Extradition should stop fugitives escaping justice, not allow British citizens to be yanked around the world on the basis of what they are alleged to have done here at home."
If Mrs May decides to allow extradition to go ahead, Mr McKinnon's lawyers are expected to launch a last-ditch application for judicial review to challenge the decision.
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