Rocket fire from Gaza into Israel continued throughout Tuesday amid reports of a massive international push for a ceasefire.
The IDF also struck six rocket launching posts belonging to Hamas across the Gaza Strip.
Prime Minister Benjamin Netanyahu told the heads of local authorities in the South that Operation Guardian of the Walls will end within several days and promised that, "fire on the Gaza border communities after the operation will elicit very fierce responses."
Egypt has proposed an Israel-Hamas ceasefire through private channels that would enter effect on Thursday at 6am, according to a report by Channel 12 news.
The report cited Palestinian sources saying Hamas has agreed to the offer, while there has been no response from Israel.
At least two people were killed and eight others wounded Tuesday afternoon when a rocket fired from Gaza made a direct hit on a packing plant at Moshav Ohad in Eshkol Regional Council.
All the casualties were foreign workers from Thailand. Two people were seriously wounded, one person was moderately hurt and the others sustained light wounds.
Some of the wounded were airlifted to Soroka Medical Center in Be'er Sheva for treatment.
Eshkol Regional Council head Gadi Yarkoni said the moshav that was hit did not have bomb shelters.
"We have been demanding the government provide adequate protection for our residents who are under constant threat from Gaza," he said.
A rocket also hit an apartment block under construction in the city of Ashdod, as massive barrages were fired at the south. There were no injuries initially reported.
The rocket fire on the South continued throughout Tuesday afternoon, with a house south of Ashkelon being hit. No casualties were reported.
Rockets also targeted Be'er Sheva, where the Iron Dome missile defense system was able to intercept the fire.
Earlier Tuesday afternoon, a 19-year-old IDF soldier was lightly wounded in heavy mortar fire from the Gaza Strip at the Erez Crossing between Israel and the Hamas-run enclave.
The military said he was preparing trucks carrying humanitarian aid into the Strip when he was hurt.
The wounded soldier was taken to Barzilai Medical Center in Ashkelon.
Residents of the Israeli communities within a 4km range of the Gaza border were ordered into bomb shelters at 1pm Tuesday until further notice.
The order came as Gaza terror groups resumed heavy rocket and mortar fire on the area following a six-hour lull.
Homes were damaged in the Eshkol Regional Council when rockets landed close by and an agricultural farm was also hit by rockets, killing some livestock.
The Kerem Shalom Crossing also came under heavy fire from Gaza as it opened to allow medical, fuel and food supplies into the Strip.
The attacks from Gaza resumed at around midday after a six-hour break, targeting communities near the border as well as the city of Ofakim some 20km from the Strip.
Israel also continued to strike targets in the Gaza Strip on Tuesday, toppling a six-story building that housed libraries and educational centers belonging to the Islamic University, leaving behind a massive mound of rebar and concrete slabs.
Israel warned the building's residents ahead of time, sending them fleeing into the predawn darkness, and there were no reports of casualties. Israel said it was targeting terror groups, their tunnels and rocket launchers across the territory.
The IDF said that since the early morning hours, it has hit nine rocket launchers, a squad responsible for firing anti-tank shells at targets close to the border and the homes of senior members of the Hamas terror group.
According to the military, in the past 24 hours it has targeted 12 homes belonging to senior Hamas commanders, including that of the deputy leader of the Northern Gaza Brigade.
Health officials in the Strip said that they had no reports of Palestinians killed overnight in the ongoing Israeli strikes on the enclave, the first apparent reduction of casualties since fighting erupted on May 10.
Greek Foreign Minister Nikos Dendias arrived in Israel on Tuesday for meetings with his Israeli counterpart Gabi Ashkenazi and Palestinian Foreign Minister Riyad al-Maliki.
The purpose of the visit was said to be an expression of solidarity with Israel as Greece hoped to prevent further violence.
Dendias was also due to meet with officials in Jordan and Egypt.
In a briefing for reporters early on Tuesday, IDF Spokesperson Brig. Gen. Hidai Zilberman said that 90 projectiles were identified to have been fired from Gaza as of Monday evening and until early on Tuesday, 20 of them landed short of the border and that Iron Dome missile defense system intercepted dozens more.
In strikes overnight, the IDF said it targeted the affluent quarter of Rimal in Gaza City and approximately 15 kilometers of the Hamas tunnel system that runs beneath the Strip as is used to transport people and weaponry.
A home in the border area was hit by a rocket on Tuesday morning, causing extensive damage. The family was not in the house at the time and there were no injuries.
Israeli forces downed a UAV (unmanned aerial vehicle) that approached Israel's border with Jordan on Tuesday, the military said, without specifying where the aircraft might have originated from. The drone was being checked for explosives.
Zilberman said that at least 160 terrorists have been killed in the fighting, 130 of them affiliated with Hamas.
More than 3,000 rockets have been fired at Israeli communities since the fighting began. Ten people have been killed by rockets fired at Israel, including two children.
At least 212 Palestinians have been killed, including 61 children, according to Gaza health officials.
Meanwhile, diplomatic efforts to end the fighting continued, with U.S. President Joe Biden telling Prime Minister Benjamin Netanyahu in a Monday night call that he supported a ceasefire, the White House said.
"The president expressed his support for a ceasefire and discussed U.S. engagement with Egypt and other partners towards that end," the White House said.
Biden "encouraged Israel to make every effort to ensure the protection of innocent civilians," the White House said.
"The President reiterated his firm support for Israel's right to defend itself against indiscriminate rocket attacks," it said.
The White House said the two leaders also "discussed progress in Israel's military operations against Hamas and other terrorist groups in Gaza."
The IDF said that at least five rockets were fired at Israel from Lebanon late on Monday but all fell short.
This was the second time rockets were fired from Lebanon since the fighting began. IDF returned fire and said that the Iranian-backed Hezbollah terror group was not behind the which it attributed to Palestinian groups in the area.
Bomb shelters were opened in communities along the northern border.
The United Nations Interim Force in Lebanon (UNIFIL) said that it was bolstering security control in coordination with the Lebanese Armed Forces after it detected rockets fire from southern Lebanon.
UNIFIL added on Twitter it has intensified patrols "to prevent any further incidents that endanger the safety of the local population and the security of southern Lebanon," while the head of the mission remains in contact with the involved parties to ensure stability in the area and deescalate existing tensions.
It added the situation in the area was now calm.
On Monday, sources in the U.S. said the Biden administration approved the potential sale of $735 million in precision-guided weapons to Israel.
The Chairman of the House Foreign Affairs Committee Gregory Meeks said he will urge the president to suspend the sale, according to a report in the Washington Post.
Meeks claimed in a virtual conference Monday discussing the violence that Congress was unaware by the weapons deal, although the White House officially advised Congress of the sale on May 5, and demanded more time for deliberations.