Israeli Defense Minister Israel Katz said Monday that Hezbollah Secretary General Naim Qassem is now a "marked target for elimination."
Writing on US social media company X, Katz vowed that Hezbollah "will pay a heavy price for the firing toward Israel."
"Naim Qassem, Hezbollah's secretary general, who decided on the firing under pressure from Iran, from now on, is a marked target for elimination," he added.
"Whoever follows in (Ali) Khamenei's path will soon find himself with him in the depths of hell alongside all those eliminated from the axis of evil," Katz said.
The military early Monday said that it has launched an "offensive battle" against Hezbollah in Lebanon.
Israeli airstrikes targeted Beirut's southern suburbs and areas in southern Lebanon, killing 31 people and wounding 149, according to Lebanon's Health Ministry.
Hezbollah said early Monday morning that it had fired missiles and drones at an Israeli military site in response to the near-daily attacks by Tel Aviv on Lebanon and the assassination of Iran's Supreme Leader Ali Khamenei.
Since Saturday morning, Israel and the United States have waged attacks against Iran that have killed hundreds of people, including Supreme Leader Khamenei and top officials.
Tehran has responded by firing missiles and drones at Israel and US bases and interests in several Gulf countries.
Washington and Tel Aviv accuse Tehran of pursuing nuclear and missile programs that threaten Israel and US allies in the region. Iran says its nuclear program is peaceful and that it does not seek to produce nuclear weapons.