Lebanon's National News Agency (NNA) said the suspect was identified as Wessam Dala. He is accused of carrying out the attack on a Shiite shrine in the Syrian capital.
A senior Hezbollah source told The National that Mr Dala, born in 2000, entered Lebanon illegally and was staying with relatives in Hay Al Salam, a Beirut suburb. The area is controlled by Iran-backed Hezbollah, a Lebanese Shiite armed group and political party.
The source said the suspect was wanted on suspicion he would carry out a terrorist attack.
"He was pursued, and when he learnt of the discovery of his whereabouts, he threw himself from the seventh floor, and he was transferred to St George Hospital, where he died," the source added, while making no reference to the attack in Damascus.
Syrian state media said on July 27 that a bomb planted in a vehicle exploded outside the Sayeda Zeinab shrine in the southern suburbs of Damascus, killing several people and wounding others.
The blast took place on the eve of the holy day of Ashura.
On the following day, ISIS claimed responsibility for the attack.
Hezbollah was the only Lebanese civil war militia allowed to keep its weapons after the conflict ended in 1990. It is a highly influential force in Lebanon, and its weapons are thought to rival the Lebanese military.
Hezbollah has been a leading ally of Syrian President Bashar Al Assad during the 12-year civil war in Syria.
The group, an enemy of Israel, is proscribed as a terrorist group by some Arab countries and most western states.
While some Lebanese laud the group as the resistance movement that ended Israel's occupation of Lebanese territory, other see it as holding a stranglehold over the country and having undue influence over state affairs.