In a statement circulated on Twitter, Al-Nusra Front said Suri and other members "were killed during a Crusader (Western) air strike... On April 3, 2016."
It said the raid had hit a training camp, but did not specify where in Syria.
On Monday, the Pentagon said the US military conducted an air raid on an Al-Nusra meeting in northwest Syria the previous day but could not confirm whether Suri had been killed.
Suri was a Syrian national and a "legacy" Al-Qaeda member who fought in Afghanistan in the late 1980s and 1990s, Pentagon press secretary Peter Cook said.
Suri's death had been reported by the Britain-based Syrian Observatory for Human Rights, which said that his son and at least 20 other jihadists were also killed Sunday in strikes on positions in Idlib province.