Hezbollah fired on Monday a volley of rockets from Lebanon at an Israeli military base in Syria's occupied Golan Heights, in response to Israel's deadly airstrikes in Lebanon's Baalbek.

Two Hezbollah members were killed in the Israeli strikes in the eastern Baalbek area, two security officials in Lebanon and a source close to Hezbollah told AFP.

During a funeral procession for a Hezbollah fighter on Monday, one of the group's lawmakers, Hassan Fadlallah, lambasted the Israeli airstrikes on Lebanon’s eastern district of Baalbek, a Hezbollah bastion near the border with Syria.

The National Moderation parliamentary bloc has launched an initiative aimed at breaking Lebanon’s presidential vote deadlock, amid a “green light” from Speaker Nabih Berri, the Nidaa al-Watan newspaper reported on Monday.

Berri has asked the Moderation bloc not to use the phrase “dialogue session” so that it does not clash with his initiative, the daily said.

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.”