Updated

Even as the death toll mounts in the Gaza Strip, attempts to broker a cease-fire between Hamas and Israel have so far run aground — in part because they have become mired in the deep divisions between Mideast countries.

At the center of the problems is the bitter enmity between Egypt and its Gulf allies like Saudi Arabia on one side and Gaza's Hamas rulers and its allies, Turkey and Qatar, on the other.

An Egyptian cease-fire proposal quickly fell apart the past week when Israel accepted it but Hamas rejected it. Hamas demanded greater guarantees for the lifting of the blockade of Gaza, enforced by Israel and Egypt. The Egyptian proposal called for both sides to halt hostilities unconditionally — dangling only a promise of further talks that could address the closure.

Qatar-based Hamas spokesman Hossam Badran described Cairo's cease-fire proposal as "all but dead," calling it a "surrender" to Israel.

He and other Hamas officials said they were instead turning to Qatar, which they said had an initiative that would address their demands, including release of prisoners and giving unfettered access to the densely populated strip. That quickly sparked accusations by Egypt that Hamas' allies were undermining it.

"The Hamas-Qatar-Turkey axis is trying to abort Egypt's role, which is the region bulwark in the face of a plot aimed at fragmenting the region into rival mini-states," Egypt's Foreign Minister Sameh Shukri told reporters Thursday night, just before Israel announced the start of its ground assault into Gaza.

Shukri said Egypt is in a "very tense and difficult" relationship with Hamas, where reaching common ground is nearly "impossible."

On Saturday, Shukri said he knows of no other initiative and that "the Egyptian initiative remains the initiative on the table" with international support.

Speaking next to visiting French Foreign Minister Laurent Fabius, Shukri said there was no intention to amend the proposal, which he said meets the demands of both sides.

The tensions are rooted in the turmoil in Egypt over the past year. Hamas spawned off the Muslim Brotherhood, which Egypt's government has branded a terrorist organization since the military's ouster of Islamist President Mohammed Morsi last summer. Egyptian authorities have been cracking down hard on Morsi's Brotherhood and accuse Hamas of helping Islamic militants waging a campaign of violence in Egypt, a claim the group denies.

Egypt also has tightened the closure on Gaza by destroying smuggling tunnels under the border that were largely propping up the strip's economy. That has thrown Hamas into a financial crisis.

Turkey and Qatar were also close allies of Morsi and the Brotherhood — and the result has been deep tensions between them and the government of Egyptian President Abdel-Fattah el-Sissi, the former army chief that ousted Morsi.

The cease-fire plan put forth by Egypt now is virtually identical to one presented by Morsi during the last round of Gaza fighting. At the time, both sides accepted the accord, and Morsi was lauded for his mediation. Now, however, there is not only deep mistrust in the way, but also increased Hamas expectations for an end to the stifling blockade.

Badran told The Associated Press that Hamas wants the permanent opening of the Rafah crossing with Egypt, an arrangement to allow Gazans to pray in Al-Aqsa mosque in Jerusalem and the release of political prisoners held by Israel.

Hamas and its allies Qatar and Turkey also are pressing for the opening of an airport and seaport in Gaza under international administration.

"We have to impose conditions and guarantees to prevent the recurring of this assault and lift this inhumane and illogical blockade on Gaza," Qatari Foreign Minister Khalid al-Attiya said in a speech last week.

Turkish Foreign Minister Ahmet Davutoglu has said the blockade must be lifted. "If a tunnel is closed, an airport has to be opened. There has to be a breathing space. The situation now is beyond a ghetto," he said in an interview Thursday with Turkey's NTV television.

However, Egypt — as well as Hamas' rival Palestinian President Mahmoud Abbas — is wary of any moves that would give Hamas breathing room and strengthen its hold on Gaza.

"There is no way Egypt is going to let the crossing open for Hamas to come and go without control," an Egyptian foreign ministry official said. He said Egypt could agree to international observers — or forces from Abbas' Palestinian Authority — controlling the crossing, as long as they are on the Gaza side.

The official spoke on condition of anonymity because he was not authorized to speak to journalists.

Sameh Seif al-Yazel, a security expert and one-time member of el-Sissi's presidential campaign, said Egypt would accept arrangements giving free movement in and out of Gaza as long as they are endorsed by the Palestinian Authority. "Egypt is siding with the Palestinian Authority and the Palestinian people, not Hamas," he said.

Abbas has backed the Egyptian proposal, saying it is identical to the 2012 cease-fire brokered by Morsi. After visiting Cairo, Abbas left to Turkey, where he is expected to meet with Hamas' top leader, Khaled Mashaal, before heading to Qatar and other Gulf countries in an attempt to bridge the gaps.

Turkey meanwhile has denied undermining Egypt's efforts. Turkish Prime Minister Recep Tayyip Erdogan angrily denounced el-Sissi, saying, he "is an oppressor, a coup leader. He is the person who has been blocking Hamas' routes."

Soon after Morsi's ouster, Egypt and Turkey recalled their ambassadors. Egypt also shut down Qatar-owned satellite news network Al-Jazeera's bureau there, accusing it of being a mouthpiece for the Brotherhood.

Throughout the current Gaza crisis, Egypt's media — which is overwhelmingly supportive of el-Sissi — has been sharply critical of Hamas. TV stations have accused Hamas of wasting Palestinian blood and "commercializing" the Palestinian cause.

The editor in chief of the state-owned Al-Ahram on Saturday accused Hamas and its allies of seeking to "stir public opinion" against el-Sissi and his government by prolonging the conflict with Israel.

In one recent program, fiercely pro-military television presenter Tawfiq Okasha described Hamas and those who show any sympathy for it as "dogs" and waved his shoes in a show of contempt for "those who don't appreciate Egypt's weight."

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Associated Press writers Suzan Fraser in Ankara, Turkey, and Abdullah Rehby in Doha, Qatar, contributed to this report.