Site Intelligence, a US militant monitoring group, cites an AQAP statement as saying Ansi was killed in April in the port city of Mukalla.
There is no US confirmation. Ansi had appeared in a number of AQAP videos.
In one, he claimed the group was behind the attack on the Paris offices of French satirical magazine Charlie Hebdo in January, in which 12 people died.
Site also said Ansi appeared in videos claiming AQAP was holding - and had then killed - the American journalist Luke Somers.
Mr Somers, and South African teacher Pierre Korkie, were killed by al-Qaeda militants in Yemen last December.
They died during a failed rescue bid by US and Yemeni special forces.
Site said Ansi's eldest son and other fighters also died in the air strike in Mukalla.
Al-Qaeda seized Mukalla, the capital of the eastern Hadramawt province, in early April, along with a large army base nearby. Dozens of Aqap members were freed from the city's prison.
But within three days, most al Qaeda troops had been forced out of the port city by local tribesmen.
In another video, Ansi called for Sunni Muslims to attack Houthi forces, who took over large parts of Yemen last September.
A Saudi-led coalition has been taking part in air strikes against Houthis for the last six weeks.
Last month, the US Secretary of Defence Ash Carter said Aqap were "making direct gains on the ground" in Yemen thanks to instability elsewhere in the country.
A US air strike in Yemen has reportedly killed Nasser al-Ansi, a top militant of al-Qaeda in the Arab Peninsula.