The report comes after Egypt and the United Nations exerted pressure on Israel to restore calm between the IDF and Hamas. In response the Israeli army carried out air strikes on around 20 Hamas targets in Gaza.
Israel has not yet responded to the Hamas demand.
The sudden burst of fighting has complicated the mission of Egyptian mediators, who have been trying to achieve calm and prevent a full-blown conflict between the militant Hamas group and Israel.
Among the targets was Hamas' new headquarters for the General Security Services, the military said.
He added, "The Islamic Jihad will abide by the ceasefire if the occupation (Israel) does the same".
Restrictions remained in place, however, limiting gatherings outdoors to 100 people and those indoors to 500.
Four Palestinians died Friday during protests that turned violent at the Gaza-Israel border, the Gaza Ministry of Health said.
Israel targeted several sites, including two weapons manufacturing facilities, five military compounds, a training compound, a tunnel and an anti-aircraft observation post.
They are also protesting over Israel's crippling blockade of Gaza.
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IDF spokesman Ronen Manelis said on Saturday that Israel held Hamas responsible for the violence and that it was expected to have control over events in the Gaza Strip.
"From our perspective, part of the address by which we will deal with this fire is also in Damascus and the Quds Force", he added.
"Over the night, we have passed those messages to the relevant parties, and I'd emphasize and say that our response is not limited geographically", Conricus said in a briefing to reporters Saturday morning.
Israel's Iron Dome rocket-defence system reportedly intercepted 13 rockets, two of which landed in Gaza and the remainder fell in open spaces in southern Israel.
Islamic Jihad, a smaller militant group, implicitly claimed responsibility for the rocket fire.
In Gaza City, a four-storey building was completely destroyed in a strike, AFP correspondents reported.
There were no reports of casualties in Gaza as a result of the Israeli air strikes, which began late Friday and continued on Saturday.
The protests along the border, which have frequently involved tens of thousands of demonstrators, began in March.
Gaza Health Ministry spokesman Ashraf al-Qidra said an Israeli airstrike damaged the Indonesia Hospital in northern Gaza.
A fifth man died east of Bureij in central Gaza when a hand grenade he was holding exploded accidentally, witnesses said.