The Al-Qassam Brigades, the armed wing of Hamas, issued a statement denouncing the Israeli assassination of two of its fighters, in northern Gaza, and holding Israel accountable for any future escalation.
"Targeting Abu-Hussein and Murtaja was a clear example of the flagrant Israeli violations against Palestinian journalists", the report said.
High level sources in Gaza told "Al-Khaleej Online" newspaper that Hamas welcomed the amended Egyptian initiative, which added to the Hamas-Fatah reconciliation process three more conditions, including a prisoner exchange deal, truce in Gaza, and ending the siege on the strip.
A video released by the Israeli military shows two men in a watchtower, said to be in northern Gaza, and puffs of smoke as one appears to open fire.
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Israel has called the demonstrations an intentional provocation, claiming Hamas is paying civilians to take part in them, making the militant organization responsible for the bloodshed. It then cuts to footage of a shell hitting the post and an explosion.
Tensions have escalated since late March, when Hamas launched what would become regular mass protests along Israel's perimeter fence with Gaza.
Israel's security cabinet discussed the proposal on Sunday, but afterwards ministers said only that the Israeli military was "prepared for any scenario".
Since Palestinian protests and clashes along the Gaza border erupted at the end of March, at least 160 Palestinians have been killed by Israeli fire.