Lebanon is studying a verbal proposal by U.S. envoy Amos Hochstein who visited the country Monday in an attempt to tamp down tensions on the Lebanese-Israeli border, caretaker Prime Minister Najib Mikati said.

"We have questions and we are waiting for answers from the U.S. envoy," Mikati told al-Jadeed in an interview on Tuesday evening.

U.S. mediator Amos Hochstein has offered Lebanon “economic incentives” during his visit this week to Beirut, a Lebanese source said.

“The incentives are related to U.S. guarantees for importing gas from Egypt and electricity from Jordan, which means exempting Lebanon from the Caesar Act sanctions imposed on Syria’s government and speeding up exploration operations in Lebanon’s waters,” the source told Kuwait’s al-Anbaa newspaper.

A family of three, a father, his wife and their son were killed Tuesday in an Israeli strike on a house in the southern border village of Houla.

The National News Agency said the three killed were civilians but Hezbollah later announced the death of the father, Hassan Hussein, and his 25-year-old son, Ali Hussein, "on the road to Jerusalem" -- the phrase used for Hezbollah fighters killed by Israel.

Israeli Defense Minister Yoav Gallant on Tuesday held talks with U.S. mediator Amos Hochstein, who had visited Lebanon on Monday as he seeks to mediate a solution for the Israel-Hezbollah border conflict.

Monday’s meeting between the National Moderation Bloc and Hezbollah’s parliamentary bloc did not lead to setting a date for a consultative parliamentary meeting that would kickstart the Moderation Bloc’s initiative that is aimed at securing the election of a new president, media reports said.

Caretaker Prime Minister Najib Mikati called Tuesday on the international community to stop Israel's agressions and to work on peaceful solutions for the region.

"In order for this region to know peace and prosperity, Israeli aggressions against southern Lebanon and Gaza must stop [...] and Israel must be compelled to implement international resolutions, especially Resolution 1701," Mikati said, adding that "Israel must also withdraw from all the Lebanese territories that are still occupied."

The head of Hezbollah’s parliamentary bloc, MP Mohammad Raad, noted Tuesday that Hezbollah is “still committed to accuracy in the deterrence equations it has imposed on the Israeli enemy.”

Parliament Speaker Nabih Berri is open to a presidential initiative by the National Moderation bloc.

"Their initiative is clear - there is no ambiguity,” Berri told al-Joumhouria, in remarks published Tuesday, adding that he has expressed "positivity" towards it.

A “positive atmosphere” engulfed the meeting that Speaker Nabih Berri held Monday with U.S. mediator Amos Hochstein, informed sources said.

Hezbollah targeted Tuesday Kiryat Shmona and carried out four other attacks on northern Israel, while Israeli artillery shelled in response the Marjayoun plain and al-Wazzani.

Hezbollah said it targeted Kiryat Shmona "in response to the Israeli attacks on southern villages and civilians, especially on Bint Jbeil."