Two judges began hearing Mike Lynch’s challenge at a High Court hearing in London on Wednesday.
Then home secretary Priti Patel approved Mr Lynch’s extradition to the United States to answer criminal fraud charges in January 2022.
A judge at Westminster Magistrates’ Court had ruled Ms Patel could decide whether to order extradition.
Mr Lynch is appealing against that ruling by District Judge Michael Snow.
Lawyers representing the US Government say Judge Snow made “correct” decisions.
Lord Justice Lewis and Mr Justice Julian Knowles are expected to consider rival arguments over two days.
Mr Lynch was at the hearing.
Ms Patel’s decision to approve extradition came after Mr Lynch lost a multibillion-dollar fraud action, at the High Court in London, over the sale of his software company Autonomy to Hewlett Packard (HP) in 2011.
Mr Lynch had been accused of deliberately overstating the value of his business before it was acquired by the American technology giant.
HP had sued Mr Lynch and Autonomy’s former chief financial officer Sushovan Hussain for about five billion US dollars (£3.7 billion) following its purchase of the Cambridge-based firm for 11.1 billion US dollars (£8.3 billion) more than a decade ago.
A High Court judge who oversaw that trial said HP had “substantially succeeded” in its various claims against the two men – but was likely to receive “substantially less” than the amount claimed in damages.
Ms Patel had wanted to consider Mr Justice Hildyard’s ruling on HP’s claim before making an extradition decision.
Mr Lynch has denied all charges against him.