Israel said it was attacking the Iran-backed group's headquarters and weapons facilities in the Lebanese capital, and Israeli and US media reported Nasrallah was the target of strikes Friday night, although a source close to Hezbollah said he was "fine".
The blasts that rocked southern Beirut Friday were the fiercest to hit the group's stronghold since Israel and Hezbollah last went to war in 2006.
An AFP photographer said dozens of buildings in the district were destroyed.
After heavy strikes sounded across the Mediterranean city Friday, Israel issued fresh warnings for people to leave part of the densely populated Dahiyeh suburbs before dawn.
Hundreds of families spent the night outside, in central Beirut's Martyrs' Square or along the seaside boardwalk area.
South Beirut resident Rihab Naseef, 56, slept outside a church.
"I didn't even pack any clothes, I never thought we would leave like this and suddenly find ourselves on the streets," Naseef told AFP.
Israel's military also announced "extensive strikes" on the Beqaa area in eastern Lebanon and on the south, saying it hit "dozens of terror targets".
It said a surface-to-surface missile fired from Lebanon fell in an open area in central Israel.
Israel's military declined to comment on Nasrallah, but claimed Saturday to have killed "Muhammad Ali Ismail, commander of Hezbollah's missile unit in southern Lebanon, as well as his deputy and "other senior officials".
The Lebanese group, which normally confirms major deaths among its ranks on the same day, has yet to comment on Nasrallah's fate.
- 'Precise strike' -
Hours earlier at the UN General Assembly, Israeli Prime Minister Benjamin Netanyahu vowed to keep fighting Hezbollah until the country's northern border with Lebanon was secured.
"Israel has every right to remove this threat and return our citizens to their homes safe," he said.
Hezbollah began low-intensity cross-border attacks a day after its Palestinian ally Hamas staged its unprecedented attack on Israel on October 7.
Israel has over the past few days shifted the focus of its operation from Gaza to Lebanon, where heavy bombing has killed more than 700 people and sparked an exodus of around 118,000 people.
Israeli military spokesperson Daniel Hagari said Friday a "precise strike" hit Hezbollah's "central headquarters" underneath residential buildings in Dahiyeh.
Lebanon's health ministry gave a preliminary toll of six dead and 91 wounded.
In the Haret Hreik neighbourhood, an AFP photographer saw craters up to five metres (16 feet) wide left by the blasts.
"I felt like the building was going to collapse on top of me," said Abir Hammoud, a teacher in her 40s.
A second wave of attacks in the same area followed early Saturday, as the Israeli military said it warned civilians to get away from three buildings in the heart of Dahiyeh.
Early Saturday, Hezbollah claimed a rocket attack on kibbutz Kabri in northern Israel, "defending Lebanon and its people".
Israel's military said sirens sounded in the north.
Hezbollah later said it launched "a salvo of Fadi-3 rockets" towards the Ramat David airbase in northern Israel.
- 'Incredibly exhausting' -
Israel this week raised the prospect of a ground operation against Hezbollah, prompting widespread international concern.
"We must avoid a regional war at all costs," UN chief Antonio Guterres told world leaders, again appealing for a ceasefire.
In Israel, too, many were weary of the violence.
"It is incredibly exhausting to be in this situation. We don't really know what's going to happen, there's talk of a ground offensive or a major operation," said student Lital Shmuelovich.
At the UN, Netanyahu also addressed the war in Gaza, saying that Israel's military would continue to fight Hamas until it achieved "total victory".
Diplomats have said efforts to end the war in Gaza were key to halting the fighting in Lebanon and bringing the region back from the brink.
"The path to diplomacy may seem difficult to see at this moment, but it is there, and in our judgement, it is necessary," US Secretary of State Antony Blinken said.
- 'Outregeous threats' -
Hamas's October 7 attack on Israel resulted in the deaths of 1,205 people, mostly civilians, according to an AFP tally based on Israeli official figures that include hostages killed in captivity.
Of the 251 hostages seized by militants, 97 are still held in Gaza, including 33 the Israeli military says are dead.
Israel's retaliatory military offensive has killed at least 41,534 people in Gaza, most of them civilians, according to figures provided by the Hamas-run territory's health ministry. The UN has described the figures as reliable.
The Lebanon violence has raised fears of a wider spillover, with Iran-backed militants across the Middle East vowing to keep fighting Israel.
Netanyahu addressed Iran in his UN General Assembly speech, saying: "I have a message for the tyrants of Tehran. If you strike us, we will strike you."
"There is no place in Iran that the long arm of Israel cannot reach, and that's true of the entire Middle East."
Iran's Foreign Minister Abbas Araghchi at the Security Council denounced what he called Netanyahu's "outrageous threats to invade other states and kill more people".
Iran's embassy in Lebanon earlier called the strikes on Beirut "a dangerous escalation that changes the rules of the game".
burs-srm/jsa
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