SayfildinTahir Sharif is accused of conspiracy to kill Americans and of supporting a terror group that took part in a 2009 suicide bombing in his native Iraq.
Five U.S. soldiers were killed when a truck filled with explosives was detonated at a military checkpoint.
Canada's justice minister granted extradition last summer, after receiving assurances from the U.S. that Sharif wouldn't face the death penalty.
But Sharif appealed that, anda hearing was held in May, and the Appeal Court reserved its decision after four hours of arguments.
Sharif, a Canadian citizen, was arrested in January 2011 at an Edmonton apartment where he lived with his girlfriend and her children.