The incident reportedly occurred in the village of al-Adisa, not far from attack tunnels dug by Hezbollah into northern Israel. The drone was apparently undamaged and returned to Israeli airspace.
A Lebanese security source told Reuters that troops fired shots from M16 assault rifles.
The Israel Defense Forces said it was "checking the report."
The incident comes less than a week after the Lebanese terror group Hezbollah said two Israeli drones rigged with explosives came down over Beirut on Sunday morning. One allegedly hit the roof of the Iranian-backed organization's media office in Dahiyeh, its stronghold in the southern part of Beirut.
The second drone, which reportedly arrived in the area 45 minutes later, exploded in the air and crashed nearby.
According to The Times of London, the Israeli drones were targeting shipping containers with "machinery to mix high-grade propellant" for precision-guided missiles.
Prime Minister Benjamin Netanyahu on Tuesday warned Hezbollah Secretary-General Hassan Nasrallah to "calm down," after he threatened to retaliate to the alleged attack.
"I heard what Nasrallah said. I suggest to Nasrallah to calm down. He knows well that Israel knows how to defend itself and to pay back its enemies," Netanyahu said.
The IDF on Tuesday ordered restrictions on the movement of military vehicles along Israel’s border with Lebanon after Nasrallah threatened to retaliate for the drones.
Meanwhile, Moscow said Tuesday that Lebanese Prime Minister Saad Hariri called Russian Foreign Minister Sergey Lavrov, asking him to use his "influence to prevent a further escalation of tensions between Israel and Lebanon."
According to Hariri's office, Hariri told Lavrov that the attack in southern Beirut was "a dangerous act and aggression" and that he's counting on Russia's role to "avoid more escalation and tension and to send message to Israel to stop violating Lebanese sovereignty."
Lebanese President Michel Aoun said Monday that his country had a right to defend itself after the drone incident, which he likened to a "declaration of war."
"What happened was similar to a declaration of war which allows us to resort to our right to defending our sovereignty," Aoun's office quoted him as saying on Twitter.
"We are a people seeking peace not war, and we don't accept anyone threatening us in any war," added Aoun, a political ally of Hezbollah.
Itay Blumenthal contributed to this report