GAZA CITY — For much of the past year, Gaza has sat in frustrated isolation as it struggled to recover from a devastating war with Israel. But for the people of the battered coastal enclave, the past week brought some welcome signs of relief.

First, Egypt opened its border crossing, allowing thousands of Palestinians to exit Gaza after being penned in for months.

Then, Egypt allowed in 8,000 tons of cement, a fraction of what the territory needs to rebuild. But the move suggested a thaw in relations between the Egyptian government and Hamas, the Palestinian group that governs Gaza.

Meanwhile, news has been emerging that Hamas has begun quiet negotiations with Israel via intermediaries to extend a truce and to ease the tight restrictions on the territory.

The shifts underscore how far Hamas has staked out its own path, bypassing its reliance on the Palestinian Authority government in Ramallah, the administrative center of the West Bank, for money and reconstruction. And they came as the Ramallah government, which has been rived by infighting, appeared close to collapse.

But while Hamas appears to be striking out on its own, partly in response to what it sees as intransigence by its rivals in Ramallah, there are concerns that its recent moves might undermine Palestinian unity and compromise the campaign for an independent state.

“There is some fear that we will have a separate state in Gaza,” said Sameer Abumdallala, an economics lecturer at Al-Azhar University in Gaza. “This is dangerous to the national Palestinian project.”

Gaza, a tiny territory that lies between Israel and Egypt’s Sinai Desert, has been governed by Hamas for nine years, the product of an enduring political rift among Palestinians.

The Palestinian Authority rules communities in the Israeli-occupied West Bank, in a government headed by President Mahmoud Abbas, also the leader of the Fatah party. Hamas governs Gaza since driving out Fatah in 2007, in a civil war that followed an attempt at a unity government after Hamas won the 2006 Palestinian legislative elections.

Many Palestinians view healing the rift and the formation of a unity government as the key to harnessing the energy of both parties and eventually achieving a Palestinian state. But neither Hamas nor Fatah has been willing to loosen its grip on territory, despite years of pledges to seek reconciliation.

A “consensus government,” with officials approved by both Hamas and Fatah, was set up last July under Prime Minister Rami Hamdallah as a part of a broader reconciliation attempt. But Mr. Hamdallah’s government was never able to assert itself in Gaza or lead a reconstruction effort there. The government is blamed, as are Mr. Abbas and Hamas, for the destruction that still pocks Gaza, and the high unemployment and poverty.

This week, officials said Mr. Hamdallah had appeared to tender his resignation to Mr. Abbas. But after negotiations, it appears that Mr. Hamdallah’s government will continue until officials can agree on the formation of a new one. It is unclear whether it would be composed of technocrats, Fatah loyalists and their allies, or Hamas supporters.

“It is clear that the division is widening and deepening instead of narrowing or lessening,” read a recent editorial in Al-Quds, a Palestinian daily, in response to the week’s events.

Mousa Abu Marzouk, a Hamas leader, has met four times in the past two months with Egyptian intelligence officials to try to repair relations with Egypt, according to Ahmed Yousef, an adviser to the leader of Hamas in Gaza, Ismail Haniyeh.

Hamas has been aligned with the Muslim Brotherhood, which briefly held power in Egypt until mid-2013, when the military seized control. The new Egyptian government set out to punish Hamas, which it accused of offering a haven to militants carrying out attacks in Egypt, and shut the main crossing into Gaza.

Before this week, it had been opened only five times this year, for brief periods.

Egypt has also blocked the hundreds of tunnels that once crisscrossed the border, which were used to bring goods into Gaza that Israel had tightly restricted. The closing of the tunnels was an economic blow to Hamas, which levied taxes on the trade, hampering its ability to pay government employees in Gaza for months.

Hamas officials said they had reassured Egypt that the organization was not offering sanctuary to Islamic extremists seeking to end the rule of President Abdel Fattah el-Sisi.

Hamas officials said the meetings had helped improve relations. The border crossing was meant to open for three days, they said, but stayed open for a week. The cement was also a good-will gesture, they added.

“Perhaps it means there is a fundamental change with their relations with Gaza,” said Mr. Yousef, the adviser, speaking of Egypt. “We are on the brink of an explosion here. It will blow up either in Gaza or on the border because of the pressure,” he said. “The border is a safety valve that releases the pressure on Gaza.”

Egyptian officials, for their part, said that policy toward Hamas in Gaza had not changed.

Hamas’s drift away from its Palestinian rivals has been bolstered by Qatar, which has long been sympathetic to conservative Islamic governments.

In April, Qatar began shipping in reconstruction materials, although it was unclear when actual building would commence. It plans to rebuild 1,000 demolished apartments and repair the two main roads that run Gaza’s 25-mile length.

The materials were being transported through Israel, which appeared to be reluctantly shoring up the Hamas government, seeing it as a lesser evil than hard-line militants, loyal to the Islamic State, who now are trying to assert power in Gaza.

Israel was “alleviating the crisis in Gaza by allowing in more materials so as to brace Hamas’s rule,” said Ofer Zalzberg of the International Crisis Group. “Hamas is still viewed as an enemy,” he said, but one that Israel finds “more useful” to help than defeat.

A spokesman for the Israeli agency that handles Palestinian civil affairs said that about 1.1 million tons of reconstruction materials had been allowed to pass into Gaza from Israel since September, but he did not provide further details. Negotiations between Israel and Hamas, facilitated by European and Arab mediators, have also begun, according to a Hamas official, who spoke on the condition of anonymity because of the matter’s sensitivity. The negotiations have been widely reported, however, in Hebrew and Arabic news media. The Hamas official said it was unclear if the talks, which have focused on the extending the truce in exchange for loosening the restrictions on Gaza, would produce any results.

Still, Gaza’s needs remain vast: Less than half of 137,000 homes damaged in the summer war have been repaired, and none of the 9,000 homes that were completely destroyed have been rebuilt yet, said Adnan Abu Hasna, a spokesman for the United Nations Relief and Works Agency. The United Nations counts only homes of registered refugees in Gaza, and the total amount may be much higher. Mr. Abu Hasna said the Qataris were working on different projects unrelated to the United Nations, leaving the hardest-hit families to repair their homes themselves.

Nor has a week’s opening of the border crossing served as a sustainable alternative for a regular passageway out of Gaza. Of the estimated 15,000 Palestinians in Gaza on official lists waiting to leave, only 3,200 were able to exit.

And whatever small advances there have been have come at the cost of what Palestinians ultimately seek: a united homeland.

“This is about the fate of the land, our state, our future and our present,” the Al-Quds editorial said, warning about the collapse of the recent government and Hamas drift. “History will have no mercy, and our people will never forget!”

Majd Al Waheidi contributed reporting from Gaza City, and Rami Nazzal from Ramallah, West Bank.

A version of this article appears in print on June 21, 2015, on page A9 of the New York edition with the headline: Gaza Welcomes Break From Strife and Reopening of Egyptian Border.