A Lebanese security official said Israel had also conducted at least six strikes on south Beirut, while Syrian state media reported deadly strikes around the capital Damascus.
Despite international calls for de-escalation, Israel has vowed to keep fighting Iran-backed Hezbollah and declared a military zone in parts of its border with Lebanon.
Israeli Defence Minister Yoav Gallant warned the battle was not over, even after a massive strike on Beirut that killed Hezbollah chief Hassan Nasrallah on Friday, dealing the group a seismic blow.
Israel's military said troops, backed by airstrikes and artillery, launched ground raids targeting Hezbollah "in villages close to the border".
The targets "pose an immediate threat to Israeli communities in northern Israel", the military said.
US Defense Secretary Lloyd Austin late Monday gave Washington's backing to Israel "dismantling attack infrastructure along the border".
Hezbollah has said it is "ready if Israel decides to enter by land".
- Deadly strikes on Damascus -
As Israel announced its ground raids, Syria's official news agency SANA said the country's air defence systems had intercepted three rounds of strikes in the Damascus area.
State television said anchor Safaa Ahmad was killed "in the Israeli aggression" on Damascus, while SANA reported three civilians killed and nine others wounded.
There was no immediate comment from the Israeli military, which has carried out hundreds of strikes on Syria in recent years.
As the Israeli air force pounded south Beirut, a Palestinian camp official in south Lebanon said an Israeli strike hit Ain al-Helweh camp in the city of Sidon, targeting a Palestinian militant.
The strike hit "the house of the son of Mounir Maqdah," the official said, requesting anonymity to discuss sensitive matters. Israel has accused Maqdah of heading the Lebanese branch of Fatah's armed wing.
Hezbollah earlier said it "targeted" Israeli troops in the border village of Shtula, with a source close to the group saying the soldiers were "right on the border".
There was no immediate comment from Hezbollah following the Israeli army's announcement of the ground raids, but the group's al-Manar television reported the Israeli statement on its Telegram channel.
US news site Axios cited two Israeli officials saying the military incursion is "targeted and limited in time and scope and is not intended to occupy southern Lebanon".
Lebanon's national army, dwarfed by Hezbollah's military power, was "repositioning" troops farther from the border, a military official told AFP.
World leaders have urged de-escalation, with UN Secretary-General Antonio Guterres's spokesman Stephane Dujarric saying: "We do not want any sort of ground invasion."
- Israel orders Beirut evacuations -
This month, Israel launched a wave of deadly air strikes aimed at Hezbollah across Lebanon, the latest of which killed 95 people on Monday, according to Lebanon's health ministry.
On Monday evening, the Israeli military ordered people in three districts of southern Beirut to evacuate.
"You are located near interests and facilities belonging to the terrorist Hezbollah group... you must evacuate the buildings immediately," Israeli military spokesman Avichay Adraee said.
AFP correspondents in the capital heard explosions and saw a flash around ground level.
Hezbollah began low-intensity strikes on Israeli troops a day after its Palestinian ally Hamas staged its unprecedented attack on Israel on October 7, which triggered Israel's devastating assault on the Gaza Strip.
The border clashes rapidly escalated this month.
- 'Everyone is afraid' -
Israel's strikes on Lebanon have killed hundreds of people over the past week and forced up to a million to flee their homes, according to Lebanese officials.
Hezbollah and other groups launched rockets, drones and missiles at Israel over the same period, causing some injuries but no deaths.
Iran has said Nasrallah's killing would bring about Israel's "destruction", though the foreign ministry said Monday that Tehran would not deploy any fighters to confront Israel.
Austin warned on Monday of "serious consequences for Iran" if Tehran directly attacks Israel.
Lebanese Prime Minister Najib Mikati called for a ceasefire based on a recent US-French proposal, urging "an end to the Israeli aggression against Lebanon".
Earlier on Monday, an Israeli strike hit a building in central Beirut, with an armed Palestinian group saying it had killed three of its members.
Central Beirut resident Kahier Bannout, 42, said it was "supposed to be a safe area -- not a war zone".
"Everyone is afraid."
Lebanon's Health Minister Firass Abiad said more than 1,000 people have been killed since September 17.
UN refugee agency chief Filippo Grandi said: "Well over 200,000 people are displaced inside Lebanon", while more than 100,000 have fled to neighbouring Syria.
- 'Little time' -
The US, Qatar and Egypt tried for months to broker such a deal, which Netanyahu's critics accused him of obstructing.
In Gaza, the civil defence authority said Tuesday that Israeli bombardment on homes killed 12 people in the central Nuseirat refugee camp. Seven others were killed in Israeli strikes on a school sheltering displaced people east of Gaza City.
A UN Satellite Centre assessment issued Monday said "two-thirds of the total structures in the Gaza Strip have sustained damage" in nearly a year of war.
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.
Israel's retaliatory military offensive has killed at least 41,615 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.
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