A Yemeni government website said al-Shihri was a Saudi national who was released by the US from detention in Guantanamo Bay in 2007.
Al-Shihri was reportedly killed in the Hadramawt area of southern Yemen.
The army has been fighting Islamic militants in the area for months.
The Yemeni ministry of defence website said al-Shihri was killed along with six other militants in an army operation, but gave no further details.
Official sources in Yemen told the BBC the death occurred in an air raid in the Wadi Ain area of Hadramawt.
Military sources, however, said it had no information of the death and refused to confirm it. But the sources did confirm that the area was subject to air raids.
The US, which has labelled AQAP as the most dangerous branch of al-Qaeda, has operated drone attacks in the region previously.
AQAP was formed in January 2009 by a merger between two regional offshoots of the international Islamist militant network in Yemen and Saudi Arabia