As Hamas struggles to maintain control in Gaza, new aid mechanisms expose its governance failures, increasing public calls for an end to the war and for hostages' return.

The Israeli military says it was looking into the reported deaths in Al-Bureij and near the American aid centre in southern Gaza.

Hamas sources said last week the group had accepted a US-backed deal, but on Thursday political bureau member Bassem Naim said the new version meant "the continuation of killing and famine... and does not meet any of our people's demands, foremost among them halting the war".

Prime Minister Benjamin Netanyahu admitted to hostages' families that one of the two reasons that Hamas is still in power is its profits from the humanitarian aid.

The Israeli military is again advancing into Gaza and has blocked most aid, causing widespread hunger. A new but widely criticized system for getting relief in started operating this week.

Hamas is warning desperate Palestinians from heading to the controversial US-backed food distribution sites in Gaza, which saw chaos erupt upon opening Tuesday as thousands of hungry Gazans

Local journalists told ABC News that thousands of people gathered at the site northeast of Rafah in the hope of receiving food aid, but there was not enough to satisfy demand when distribution began.

A U.S.- and Israeli-backed private organization is overhauling the aid distribution system in Gaza. The United Nations says it undermines humanitarian principles.

Israel is facing growing international pressure as aid distribution efforts in Gaza devolved into chaos during the first days of a controversial new mechanism backed by the United States and Israel.

Palestinian militant group Hamas on Thursday said it was examining a new deal proposed by a US envoy, as rescuers said at least 44 people were killed in Israeli attacks across the Gaza Strip.Negotiations on a ceasefire to end 18 months of devastating war have yet to yield a breakthrough since Israel resumed operations in the war-ravaged territory in March after a brief truce.