Top Hezbollah commander Mustafa Badreddine was killed in Syria by artillery shells fired by Sunni Islamist insurgents, according to the Shia Muslim group.
A war monitoring group cast doubt on the claims, saying there had been no shelling by insurgent groups in that area for more than a week.
Hezbollah is fighting in Syria, backing the government of President Bashar al-Assad against a range of Sunni groups including the so-called Islamic State and the al Qaeda-affiliated Nusra Front.
Damascus airport and its surroundings are controlled by the Syrian government and allied forces.
Between it and government-held central Damascus, rebels control a portion of the Eastern Ghouta suburb, which has experienced fighting for most of the conflict now in its sixth year.
"There has been no recorded shelling or firing from the Eastern Ghouta area onto Damascus International Airport for more than a week," Syrian Observatory for Human Rights director Rami Abdulrahman told Reuters.
Hezbollah's statement did not say when the attack took place or when Badreddine died.
Announcing his death yesterday, Hezbollah quoted Badreddine as having said he would return from Syria victorious or as a martyr.
Lebanese TV station al-Mayadeen had reported he had been killed in an Israeli attack.
Badreddine was sentenced to death in Kuwait for his role in bomb attacks there in 1983.
He escaped from prison in Kuwait after Iraq, under the leadership of Saddam Hussein, invaded the country in 1990.
His release from jail in Kuwait was one of the demands made by the hijackers of a TWA flight in 1985, and of the hijackers of a Kuwait Airways flight in 1988.
For years, Badreddine masterminded military operations against Israel from Lebanon and overseas and managed to escape capture by Arab and Western governments.
He was also one of five Hezbollah members indicted by the UN-backed Special Tribunal for Lebanon in the 2005 killing of statesman Rafik al-Hariri, one of Lebanon's most prominent Sunni Muslim figures.
Hezbollah denied any involvement and said the charges were politically motivated.
