The Israeli air force launched further air strikes against Lebanese border villages on Tuesday.
Lebanese security sources said Israeli aircraft hit Bayyada and Mansoureh.
Israeli jets also struck eastern Lebanon and roads leading to Syria.
Hezbollah fighters said in a statement that they had beaten back some of the advances in southern Lebanon.
“The Israeli forces started from Monday night until the early hours of Tuesday morning to try to take control of the triangle of Aita al-Chaab/al-Qaozah/Ramie villages but Hezbollah fighters confronted them … forcing them to retreat,” the statement said.
Aita al-Chaab is near the Lebanese border from where Hezbollah fighters entered Israel on July 12 and kidnapped two Israeli soldiers and killed eight others.
The movement said Israeli troops were trying to take control of a hill overlooking all three villages.
Hezbollah has also said it has destroyed an Israeli warship off the coast of Tyre. Israel has denied the claim.
The Israeli army said it had killed 20 Hezbollah fighters in southern Lebanon over the past 48 hours.
An army spokeswoman said: “We have killed 20 Hezbollah terrorists during the past two days in the sectors of Taibe and Al-Adeissa.”
Hezbollah said it had lost three men during the fighting.
Arab TV channel Al-Arabiya reported that three Israeli soldiers had been killed in the area.
A senior Israeli official has said the Israeli army will move deeper into southern Lebanon and hold on to that territory for several weeks, until a multinational force can deploy there.
Israel Radio, Israel Army Radio and a Israeli goverment official said ground forces would reach the Litani River, about 30km north of the Israeli-Lebanese border.
Ephraim Sneh, a senior Labour Party politician, indirectly confirmed the planned push to Litani in an interview.
Asked by Israel Radio how long troops would hold on to territory up to the Litani, Sneh said: “We are not talking about days we are talking about longer, but not about months.”
Sneh, a former deputy defence minister, spoke hours after Israel’s security cabinet approved a broader ground offensive into southern Lebanon.
However, two other government officials said the cabinet approved taking only a smaller area of land, a strip about seven kilometres from the border.