"We announce a ceasefire with Israel after Egyptian contacts with us", Dawoud Shihab, Islamic Jihad spokesperson, told Anadolu Agency.
He added, "The Islamic Jihad will abide by the ceasefire if the occupation (Israel) does the same".
"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.
Israeli aircraft have struck dozens of militant sites across the Gaza Strip as militants fired some 30 rockets into Israel.
The Israeli army carried out scores of air strikes across the besieged Gaza Strip in the early hours of Saturday, in the highest-intensity offensive on the coastal Palestinian territory since the summer.
Islamic Jihad, a smaller militant group, implicitly claimed responsibility for the rocket fire.
A delegation from the Egyptian intelligence service that has been shuttling between Gaza and Ramallah and which has also been meeting with Israeli officials achieved an understanding regarding a halt to the violence by Palestinian demonstrators in Gaza along the border fence with Israel, in exchange for an easing of the Israeli blockade of the Gaza Strip, according to the report.
Israel and Hamas have fought three wars since 2008 and much of the worldwide community considers the Islamist movement a terrorist organisation.
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The victims were part of the protesters of the March of Return - done every Friday since March 30, when they were attacked by the Israeli forces.
Egypt wants to restore calm in order to pursue the broader goal of Palestinian reconciliation between Hamas and the West Bank-based administration of President Mahmoud Abbas. According to an Israeli army statement, 16,000 Palestinians gathered along the Gaza border.
Israel accuses Hamas of using the large protests as cover to stage border infiltrations and attacks. A fifth protester succumbed to his wound Saturday, the Health Ministry said.
Israel rarely acknowledges it has reached agreements with terror groups in Gaza.
The demonstrations, named the Great March of Return, call for the return of refugees to their homes and lands from which they were displaced in 1948. Widespread desperation among Gaza's 2 million residents has fueled the participation in the Hamas-orchestrated protests.
Israel has threatened to expand its strikes to Iranian forces in Syria, which could lead to a confrontation with Iranian and Hezbollah fighters.
Israel has struck Syria dozens of times in recent years, saying it is preventing Iran from supplying advanced weapons to enemies of the Jewish state.
Peace talks between Israel and the Palestinians broke down in 2014.