Israeli strikes near the Hezbollah-dominated city of Baalbek killed two group members Monday, two security sources told AFP, in the first strikes on Lebanon's east since clashes began after the Gaza war.

"Two Hezbollah members were killed in the Israeli strikes near Baalbek," a security official told AFP, with another security source also confirming the toll.

An Israeli airstrike targeted Monday a car in the Tyre district town of Majadel, killing at least one person, local rescuers said, without providing further details.

Israeli airstrikes had targeted earlier on Monday other areas deep inside Lebanon including Bouday near the northeastern city of Baalbek and Iqlim al-Tuffah, after Hezbollah shot down a "large" Israeli drone there.

Lebanon will respond next week to a French proposal for halting the Israel-Hezbollah cross-border clashes, caretaker Foreign Minister Abdallah Bou Habib said on Monday.

Israeli Defense Minister Yoav Gallant has said that Israel would increase its strikes on Hezbollah even if a ceasefire is reached in the Gaza Strip.

“We are planning to increase the firepower against Hezbollah, which is unable to find replacements for the commanders we are eliminating,” Gallant said during a visit to the Israeli army’s Northern Command headquarters in Safad.

The U.S., Israel and Lebanon do not support a French proposal for a truce between Hezbollah and Israel, prominent sources said.

The sources told Asharq al-Awsat newspaper, in remarks published Monday, that Israel prefers that U.S. special envoy Amos Hochstein continue his mediation.

U.S. Ambassador Lisa Johnson has visited the new American Corner in Beirut for its official opening Saturday, the U.S. Embassy said Monday.

The American Corner offers, all free of charge, access to information about American culture and values, educational opportunities in the United States, English language training, life skills and cultural activities. These services are available to all Lebanese, regardless of background.  

Hezbollah announced launching two fresh attacks on Israeli military posts on Sunday morning.

In a statement, the group said it targeted with missiles a gathering of Israeli troops near the Ramim barracks, achieving “direct hits.”

A court in Cyprus has ordered two men to remain in police custody for six days on suspicion of people smuggling. The men were identified as the drivers of two boats that brought 146 Syrian refugees and one Lebanese migrant to the east Mediterranean island nation.

Police said the suspects are Lebanese nationals aged 19 and 21.

Hezbollah on Saturday said it fired two heavy-caliber Burkan rockets at a gathering of Israeli troops on the Cobra Hill in northern Israel, achieving a “direct hit.”

It later said that it targeted a group of Israeli soldiers in the vicinity of the Dhayra military post, also achieving a "direct hit."

The health ministry condemned Friday the "direct" attack on a civilian health center, after two paramedics affiliated with Hezbollah and one of the group's fighters were killed in an Israeli strike on the southern border town of Blida.

The ministry called on the international community to hold Israel to account for its "dangerous violations".