The fugitive, known as Meshal al-Shimmari, was sentenced in December for belonging to the al-Qaida-linked Peninsula Lions group, which staged several bloody attacks in Kuwait last year.
The sources said Shimmari, 35, showed up at the Court of Appeals on Wednesday.
A former Kuwaiti army conscript, he is one of several stateless Arabs who were part of a group of 37 Islamists tried for belonging to Osama bin Laden's al-Qaida in Iraq and Saudi Arabia.
According to the sources, Shimmari was a low-level member of the Peninsula Lions group, which Kuwaiti authorities said were involved in four gun battles in January 2005 that killed nine Islamists and four members of the security forces.
Six members of the group were sentenced to death at the trial for belonging to an "extremist" group, calling for attacks on state facilities and trying to kill Kuwaiti security forces and foreign forces in the country.
Kuwait, which hosts up to 30,000 US troops, has cracked down on Islamists opposing the foreign military presence there. Diplomats say radical Islam is taking hold among Kuwaiti youth.
Neighbouring Saudi Arabia and Iraq are also battling al-Qaida militants bent on destabilising their governments