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.