The US wants to deport Glasgow-born Gary McKinnon from the UK and it has gained the approval of former home secretary John Reid on its extradition request.
McKinnon, a former sys admin, says he spent two years hacking into a number of US government systems looking for evidence of extra-terrestrial life. He is accused of illegally accessing computers belonging to the US army, navy, air force, NASA and the Department of Defense.
McKinnon is asking the Lords to overturn the proposed extradition on the grounds of an abuse of legal processes by US officials engaging in plea bargaining.
Representing McKinnon in the House of Lords on Monday, lawyer David Pannick told Law Lords that prosecutors said McKinnon faced 8-10 years in jail if he contested the charges, but only 37-46 months if he co-operated.
McKinnon claims that, as part of the offer if he co-operated, he was told that he could serve part of his sentence in the UK.
He claims that the US was not prepared to guarantee the offer in writing, and he declined to co-operate.
Pannick argued on Monday that it was not disputed that the courts could refuse to extradite people if they considered there had been an abuse of process.
He said: "If the United States wish to use the processes of English courts to secure the extradition of an alleged offender then they must play by our rules."
Clare Montgomery, a lawyer representing the Home Office, said no threats were issued by the US authorities and the extradition should be allowed to proceed.
The Law Lords have the right to block the extradition, in which case McKinnon could still face charges in the UK.
McKinnon's case has already been through multiple legal stages. Last year, his lawyers told London's High Court that he had been subject to improper threats and that extradition would breach his human rights.
McKinnon says that if he loses the Lords appeal, he will take the case to the European Court of Human Rights.
If the extradition goes ahead, McKinnon faces a sentence of up to 60 years under terrorism charges.
If he wins the case, McKinnon says he will "spend a lot of time" trying to get the Extradition Treaty changed. US Congress has not ratified the Treaty, which was created in 2003.
The law lords will present their verdict in around three weeks.