Israel's war cabinet was due to meet later on Tuesday mainly to discuss the northern front, an Israeli official said.
Israeli government spokesperson David Mencer said the fighting in the area was "not a sustainable reality", adding that Israel was committed to ensuring the return home of tens of thousands of Israelis evacuated from the north.
"It is up to Hezbollah to decide if this can be accomplished by diplomatic means or by force," he said. "We are defending this country and no one should be surprised by our response."
Amos Hochstein, a senior adviser to U.S. President Joe Biden at the heart of diplomatic efforts seeking de-escalation, said last week a land border agreement between Israel and Lebanon implemented in phases could dampen the conflict.
Israel has gone to war numerous times in Lebanon.
"WE MUST... DESTROY THEM"
Earlier on Tuesday, Israeli far-right ministers Bezalel Smotrich and Itamar Ben-Gvir both urged more military action.
"There cannot be peace in Lebanon while our land is hit and people here are evacuated,"
Ben-Gvir said following a tour in the northern city of Kiryat Shmona in a video statement shared on X. "They're setting fires here, we must burn all of Hezbollah's strongholds, destroy them. War!"
Ben-Gvir and Smotrich are members of Israel's security cabinet but not of the war cabinet.
The violence, which has oscillated for months, escalated in recent days. Hezbollah announced on Tuesday it had launched a squadron of one-way attack drones at an Israeli military barracks for a second consecutive day, calling it a response to a deadly Israeli attack in Naqoura, Lebanon.
Sirens sounded in northern Israel, where rockets fired from Lebanon set off wildfires on Monday.
The Israeli military said a "suspicious aerial target" that crossed from Lebanon fell in the Mount Hermon area, with no injuries reported.
Israeli warplanes struck a Hezbollah member in Naqoura, as well as two other locations in southern Lebanon. Israeli artillery struck five locations in the south, the Israeli military said.
Israeli strikes have killed some 300 members of Hezbollah since Oct. 7, and around 80 civilians. Attacks from Lebanon on Israel have killed 18 Israeli soldiers and 10 civilians, the Israeli military says.
On Monday, Hezbollah said it launched a squadron of drones against an Israeli target in response to Israel's killing of a Hezbollah member. While Hezbollah has used drones in the conflict, it marked the first time it had declared launching a squadron of them.