Egypt said Monday that it preferred that the Palestinian Authority president, Mahmoud Abbas, take control of the breached border between Gaza and Egypt, seeming to exclude Hamas, the Islamist group that took control of Gaza from Mr. Abbas in June.

The Egyptian foreign minister, Ahmed Aboul Gheit, told the United States and the European Union that Israel should cooperate with efforts to control border crossings “through the deployment of the Palestinian Authority” and “European Union monitors,” the ministry said in a statement. Mr. Aboul Gheit emphasized that Egypt would “carry out a gradual control of the Egyptian border with the Gaza Strip and bring the situation back to an acceptable condition,” the statement said.

Hamas blew up the Israeli-built wall between Gaza and the Egyptian border early Wednesday after Israel had sealed off Gaza to try to stop rocket and mortar fire into Israel. Since then, hundreds of thousands of Palestinians have traveled into and out of the northern Sinai, buying food, medicine, consumer goods and livestock.

Egypt has slowly tried to resecure the border, refusing to allow the resupply of goods to the border towns of El Arish and Egyptian Rafah and taking steps to narrow the breaches. Shops in El Arish and Rafah have largely run out of goods to sell, and tempers between Palestinians and Egyptians have begun to fray.

Egypt appears to want to return the Rafah crossing to its situation before June, when European Union officials monitored the border and Israel kept watch by video link. But even then security problems often kept the Rafah crossing shut. And Israel regularly objected to Egyptian decisions to let certain groups of Gazans, mostly allied with Hamas, come and go, sometimes smuggling in large amounts of cash.

Hamas, which considers itself the legitimate Palestinian government because of its electoral victory in January 2006, has made it clear that it wants a role in operating the crossing and calls the old arrangement “a piece of history.” Israel also regards the old arrangement as problematic, given Hamas’s control over Gaza, and wants Egypt to do more to stop the movement of militants and weapons.

Since Hamas routed Fatah forces loyal to Mr. Abbas in early June, the Rafah crossing had been shut, but Egypt seems unwilling to shut it completely again. At other crossings, Israel has allowed in only bare necessities, and it has been cutting fuel supplies to ensure that Gazan life is uncomfortable at best so long as militants are firing rockets into Israel.

But Israel’s effort to seal off Gaza backfired with the border breach, and on Sunday the government agreed to resume sharply limited supplies of gasoline and diesel to Gaza, as well as industrial diesel for the Gaza power plant. Although Israel resumed some supplies last week, the plant was producing only 45 megawatts a day, compared with its capacity of 80 megawatts, meaning continued rolling power cuts. At the same time, Israel said it intended to cut about 1.5 megawatts of electricity from the 120 megawatts provided from Israel beginning Feb. 7.

The Gaza border chaos has been overshadowed in Israel by the pending final report of the Winograd Commission into the 2006 Lebanon war and speculation about its impact on the Israeli prime minister, Ehud Olmert. A preliminary report about the first few days of Israel’s war against Hezbollah, the Lebanese militia, was fiercely critical of Mr. Olmert, the army and the defense minister at the time, Amir Peretz. The army has a new commander and has worked to learn the lessons of the war, and Mr. Peretz is gone, but Mr. Olmert has survived.

The final report, to be released Wednesday evening, is likely to be even more critical, focusing on the last days of the war, when 33 Israelis died in a final, aborted push into southern Lebanon before a cease-fire mandated by the United Nations.

Mr. Olmert has argued that the last offensive was required to pressure the United Nations Security Council into passing a resolution more favorable to Israel, but the American ambassador to the United Nations at the time, John R. Bolton, has said that the offensive did not matter. Mr. Olmert has insisted to friends that he will neither resign nor be fired.

Ehud Barak, the defense minister, who leads the Labor Party, will be a key participant, carefully watching public reaction. If he pulls out of the coalition, the government will fall. But few in the government want new elections, which currently would favor Benjamin Netanyahu of the Likud Party.

Mr. Barak may instead encourage a coup inside Mr. Olmert’s party, so there would be a new prime minister without new elections. But it is considered now more likely that because Mr. Olmert is in peace talks with Mr. Abbas, which Labor favors, Mr. Barak may give the talks a chance and call for new elections to be held in the spring of 2009.