Hezbollah, meanwhile, said it saw no sign of Israeli forces and its troops were ready to confront them.
Israel said the incursion would be focused on the narrow strip of land just across the border. But it also issued evacuation warnings covering a wider swath of Lebanon, raising fears that a large-scale ground invasion was soon to come.
In recent days, a wave of Israeli airstrikes has killed Hezbollah leader Hassan Nasrallah and several of his top commanders, while driving hundreds of thousands of Lebanese from their homes. Israel says a ground operation is now necessary to return tens of thousands of displaced Israelis to their homes in the north, pummeled by continuous rocket fire from Hezbollah since Oct. 8.
Here's what we know:
The military says that Israeli troops entered Lebanon late Monday. A military official who spoke on the condition of anonymity in line with military regulations declined to say whether troops were rotating in and out of the country or were stationed there for the foreseeable future.
Israel also said Tuesday that its forces have been operating in Lebanon for the last year, carrying out dozens of small ground operations. Military spokesperson Daniel Hagari said the current raid is an expansion of its activities over the past year, without providing precise details.
The Lebanese army and UNIFIL, a U.N. peacekeeping force stationed in the zone that Israeli forces purportedly entered, have not confirmed that Israeli troops crossed the border, although UNIFIL said it was notified that they were going to.
The military official said that at this point Israeli ground troops are "within walking distance" of the border, targeting the small Lebanese villages hundreds of meters (yards) from Israeli territory.
The military says Hezbollah militants remain in the area, despite heavy Israeli bombardment over the past few weeks. It says they are using the areas to launch attacks on Israel and to store weapons.
There were no signs of ground combat late Monday and early Tuesday in southern Lebanon.
The military official said there have been no clashes between Hezbollah militants and Israeli troops so far.
Meanwhile, cross-border fire continued. Hezbollah announced in statements that it had targeted groups of soldiers in several Israeli border areas with artillery shelling and rockets. Israel said no soldiers were injured and the targeting was unsuccessful.
Still, Israeli artillery units pummeled targets in southern Lebanon through the night and the sounds of airstrikes were heard throughout Beirut.
Hezbollah fired a rare volley of rockets toward central Israel on Tuesday, injuring one man, Israeli paramedics said.
Israel has not given a timetable for how long the incursion will last and has declined to specify how far troops will go.
The military official said that marching to Beirut, as Israel did in its 1982 invasion of Lebanon, is "not on the table." At the time, the Israeli invasion was also pitched as a limited incursion to push the Palestine Liberation Organization back.
He added that the operation in its current stages does not mirror Israel's ground incursions in Gaza, where the military entered cities with heavy manpower, artillery and tanks.
That could change, depending on whether Israel's government decides to launch a more extensive ground operation. Chris Coyle, a resident of Israel's north, told AP on Monday that Israeli tanks had been positioned along the border.
Troops that entered Lebanon are from the 98th division, the military said. The division is responsible for some of the heaviest fighting inside Gaza and includes elite units specializing in attacks behind enemy lines.
Meanwhile, Israel is expanding its evacuation warnings in southern Lebanon, sending hundreds of thousands of Lebanese fleeing from the south.
On Tuesday, the Israeli military's Arabic-language spokesperson asked residents living in villages north of a U.N.-declared buffer zone to flee. Under a U.N. resolution that ended the 2006 war, the zone was supposed to be controlled by a U.N. peacekeeping force and the Lebanese military.