On Sunday, when the Israeli security cabinet convened to discuss the Gaza truce ideas, one member - Intelligence Minister Yisrael Katz - tweeted that these should include "bringing Egyptian infrastructure, at sea and on land, to bear for the good of Gaza, under worldwide supervision".
The papers added that an Israeli drone fired a missile on Saturday at a group of Palestinians near the Gaza-Israel border fence to the east of Jabalia town, in the northern Gaza Strip. One Israeli soldier was killed by a Gaza sniper during this period.
The UN and Egypt are now trying to secure a deal between Israel and Hamas for a lasting truce.
Israel responded on July 9 by shuttering Gaza's main commercial terminal and limiting a Palestinian fishing zone off the enclave, measures it offered to reverse on Sunday.
A truck parks next to a security barrier inside the Kerem Shalom border crossing terminal between Israel and Gaza Strip January 16, 2018.
The deputy head of Hamas's politburo, Saleh al-Arouri, who served several Israeli jail terms for terrorism, arrived in the Gaza Strip late Thursday with other Hamas leaders for talks focused on renewed reconciliation efforts with Fatah and to raise the truce prospects with the terror group's Gazan leadership, Hamas-linked media reported.
However, an Israeli official told Channel 10 on Friday that Israel would not be willing to accept any long-term deal with Hamas that did not include the return of the held Israelis.
Israel withdrew its troops and settlers from Gaza in 2005, but maintains tight control of its land and sea borders.
A spike in tensions between Israel and Gaza's Islamist rulers in recent months prompted Egypt and the United Nations peace envoy Nickolay Mladenov to propose terms for a broad agreement between Israel and Hamas. A spokesman said any deal risked ending hopes of achieving Palestinian statehood.
The Tel Aviv regime may agree to a limited ceasefire with Palestinian resistance groups in the besieged Gaza Strip, Israeli officials have hinted.
But after a meeting of the representatives of Palestinian factions, including Fatah, Badran said, "There will be no state in Gaza and no state without Gaza". Twenty-four of slain protesters were minors, the ministry has said. Two other IDF soldiers also died in the incident.