Hezbollah supporters have been blocking the road to the country's only airport for two consecutive nights over a decision barring two Iranian planes from landing in Beirut.
A military statement shortly before midnight said Hezbollah had fired around 80 projectiles at the country Saturday.
The move suggests Israel may seek to strike Hezbollah for the first time from further east along Lebanon's border, at the same time creating a secure area from which it can freely reconnoitre the armed group and prevent infiltration, the sources said.
Hezbollah said its "fighters clashed with" Israeli troops Tuesday who were trying to infiltrate on the outskirts of Rab Tlatin village.
Sirens sounded in northern Israel on Wednesday, the Israeli military said, adding three Israeli military personnel were severely injured on Tuesday and Wednesday during combat in southern Lebanon.
Israel's military said the air force was carrying out extensive bombings of Hezbollah targets in south Lebanon and two Israeli soldiers were killed, taking the Israeli military death toll inside Lebanon to 11.
Hezbollah has not yet commented on Kaouk's fate, but supporters have been posting mourning messages for him since Saturday.
Nasrallah's killing in a command HQ on Friday came barely a week after Israel's deadly detonation of hundreds of booby-trapped pagers and radios. It was the culmination of a rapid succession of strikes that have eliminated half of Hezbollah's leadership council and decimated its top military command.
Fighting has escalated this week. Israel killed another top Hezbollah commander, Ibrahim Qubaisi, on Tuesday. For its part, Hezbollah has shown its capacity to continue operations, firing hundreds of rockets towards Israel in ever deeper attacks.
Nasrallah and Hezbollah have thrived with an image of invincibility after confronting Israel in 2006.
Expectations of an escalation between the two sides had risen since a missile strike in the Israeli-occupied Golan Heights last month killed 12 youngsters and the Israeli military assassinated a senior Hezbollah commander in Beirut in response.
An official from Iran-backed Lebanese armed group Hezbollah said on Saturday the group had no additional information on the stabbing attack against novelist Salman Rushdie.
Lebanon's Prime Minister Saad al-Hariri resigns citing an assassination plot against him and accusing Iran and its Lebanese ally Hezbollah of sowing strife in the Arab world.
The man believed to be Hezbollah's most senior military commander in Syria's war is killed in Damascus.
Hezbollah's most senior military commander in Syria's war is killed in Damascus, the Lebanon-based militant organisation says.
The US military has delivered more than 45 tonnes of ammunition to rebels fighting the jihadist group Islamic State (IS) in north-eastern Syria.
Hezbollah leader Hassan Nasrallah says his Lebanese Shia group will launch an attack inside Syria on Sunni militants fighting for the al-Nusra Front.
According to Israeli military assessment, Israel could suffer hundreds of civilian dead if it fights another war with Hezbollah.
