The attack was carried out between the villages of Al-Assad and Al-Dimas.
Sites in or near the Damascus International Airport were reportedly targeted in the strikes.
The 23-year-old from Syria's al-Tal region entered Lebanon illegally and settled with his relatives in Al-Salam, a southern suburb of Beirut, the security source said.
Over 10 hours after the explosions were reported, Syrian state media still had not commented on the nature of the explosions.
Four other Syrian soldiers were also wounded in the attack.
Turkey wants to normalize ties with Syria, but in order to do this it needs to see Damascus cater to some of Ankara’s demands.
The last Israeli airstrikes to hit Syria were reported about two weeks ago, when the IDF struck a Syrian air defense battery after a Syrian air defense missile exploded mid-air over southern Israel.
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