Luke Somers was shot by his captors during a raid and later died, a US official told the New York Times.
His sister, Lucy Somers, told the Associated Press that she had been notified by the FBI of his death.
UK-born Mr Somers, who was kidnapped in Yemen in 2013, had appeared in a video appealing for help.
Yemen's defence ministry confirmed a "major operation" had taken place in Yemen's southern Shabwa province on Saturday.
A drone strike believed to have been carried out by the US is reported to have killed nine suspected al-Qaeda militants in the region.
Mr Somers, 33, worked as a journalist and photographer for local news organisations. His material appeared on international news outlets, including the BBC News website.
The video of him released this week also shows a member of al-Qaeda in the Arab Peninsula (AQAP) threatening to kill Mr Somers unless unspecified demands are met.
AQAP is regarded by the US as one of the deadliest offshoots of al-Qaeda.
The group is based in eastern Yemen and has built up support amid the unrest which has beset the impoverished country since the overthrow of President Ali Abdullah Saleh in 2011.
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