Israel's Security Cabinet met on Sunday to consider Gaza proposals.
Palestinian protesters gather as tear gas canisters are launched by Israeli forces during a demonstration along the border between Israel and the Gaza strip, east of Gaza city in August 2018.
Haaretz newspaper reported Monday that Israel and Egypt have sidelined the Palestinian Authority (PA) during the talks.
One Israeli soldier was shot dead by a Palestinian sniper. Hamas and other resistance groups have linked their fate to Israel freeing Palestinian security detainees.
Hamas, which convened several of its top leaders to Gaza last week for consultations, also sounded circumspect on Sunday.
Israel's Defence Ministry described the barrier as "an impenetrable breakwater".
Australian golfer Jarrod Lyle dies aged 36 after cancer battle
'He asked that I provide a simple message: 'Thanks for your support, it meant the world. He is survived by his wife Briony , and daughters Lusi, six, and Jemma, two.
"Hamas has conducted internal meetings that have not yet ended", one senior official, Hussam Badran, told a Gaza radio station.
An additional fence will surround the large and imposing sea blockade. Egypt also restricts movement in and out of Gaza on its border.
The Palestinians and the United Nations similarly dispute Israel's assertion that it ended its occupation of Gaza in 2005.
In an interview with Ynet, Elkin added that "Hamas wants to go back in time, three months back, to a reality of calm and quiet in the Gaza Strip, we are willing to do that".
Al-Hayat al-Jadida and al-Ayyam said that President Abbas welcomed the Iraqi national football team in the presidential headquarters in the city of Ramallah, the occupied West Bank.
Reports in the Lebanese news paper Al-Akhbar earlier on Friday said the deal stipulated that Hamas must commit to "the end of the provocations along the border, or in other words, the phenomena of the flaming kites and ballons, border crossing operations and setting fire to border posts". Palestinian medics said four people were wounded.
With borders to both Egypt and Israel largely sealed in recent years, Gaza suffers from desperately high rates of poverty and unemployment.