U.S. optimism on reaching a Lebanese-Israel ceasefire deal has reached its highest level in a year, a Lebanese government source told Al-Jazeera on Monday.

The Israeli army has started decreasing military reinforcements in the Upper Galilee to pave the way for returning the displaced residents to north Israel, Saudi Arabia’s Asharq al-Awsat newspaper has reported.

The European Union “continues to mobilize all the tools at its disposal to support the people affected by the ongoing conflict in Lebanon,” the EU Delegation to Lebanon said.

The Lebanese response to the U.S. ceasefire proposal has entered the final formulation phase and is supposed to be finalized today, Monday before being sent to U.S. mediator Amos Hochstein through the U.S. Embassy in Beirut, highly informed sources said.

The Israeli military said Hezbollah fired around 100 projectiles from Lebanon into northern Israel on Monday, with Israel's air defense system intercepting some of them.

Israeli firefighters said a rocket strike on a building in the northern Israeli town of Shfaram killed a woman, while first responders reported 10 mildly injured people.

Lebanon's health ministry said Monday 10 people died in Israeli strikes on central Beirut a day earlier, after Hezbollah said four members were killed alongside its spokesman in one attack.

Israel has been heavily bombing Beirut's southern suburbs, a Hezbollah stronghold, since all-out war erupted on September 23, but attacks on central Beirut have been rarer.

Israeli strikes Sunday have killed 11 people and wounded dozens more in Lebanon’s southern Tyre region, the Lebanese health ministry said.

"The raids of the Israeli enemy on villages in the Tyre district have left 11 dead and 48 wounded," said a ministry statement.

Schools in Beirut were closed on Monday after Israeli strikes on the Lebanese capital killed six people including Hezbollah's spokesman.

Education Minister Abbas Halabi announced Sunday that schools and higher educational institutions in Beirut and surrounding areas would close for two days, after Israeli strikes hit the city center.

An Israeli airstrike hit a computer shop in central Beirut's Mar Elias area on Sunday evening, killing two people and wounding 13, the Health Ministry said.

Al-Arabiya television identified the target of the strike as "Hezbollah military commander Mahmoud Madi," saying the computer shop is owned by his brother.

U.S. mediator Amos Hochstein will visit Beirut on Tuesday and Israel on Wednesday as part of his efforts to push the ceasefire deal forward and amid reported progress in the negotiations, media reports said.