The project engineer said he paid $5,500 for 10 tons of cement, quintuple the regular price, because of scarcity caused by Israeli import restrictions and the closing of smuggling tunnels from Egypt.

In another independent initiative, the little-known Arab and International Commission to Build Gaza has cleared away rubble and is beginning to shore up columns of an Interior Ministry building erected by the former Hamas government. The commission has a Hamas official on its board, features an Egyptian dissident based in Qatar on its website, and asked on Facebook for donations to be filtered through the Islamic Bank in Tripoli.

Mohammed Abu Aqleen, the commission’s manager in Gaza, said that the Interior Ministry building was selected only because it was in danger of collapsing, and that the group had also begun to rehabilitate 20 homes throughout Gaza and to manufacture hundreds of trailers to temporarily house the displaced. He would not say where the $1 million raised so far had come from.

Referring to the effort by the Palestinian Authority and United Nations to provide assurances to Israel that so-called dual-use materials will not go to rebuild tunnels that militants used to penetrate its territory, Mr. Abu Aqleen said, “This mechanism will be just words on paper.”

But most of Gaza must await the monitoring mechanism, which Israeli and Palestinian delegations are expected to discuss further next week in Cairo. Officials at the United Nations Relief and Works Agency, the United Nations Development Program and the Norwegian Refugee Council, a leading international aid group, said the timetable remained elusive.

“It all depends on how fast the material can come in, which I just don’t know,” said Scott Anderson, Gaza’s deputy director of the relief and works agency. Asked why the material that already arrived was sitting in warehouses, Mr. Anderson said: “That I can’t answer. I’ve asked the question also.”

Some 39,000 Gazans remain in 18 United Nations schools serving as shelters, and many more are bunking with relatives. The relief and works agency has given 1,000 families a total of $1 million for temporary rental assistance or minor repairs. The housing ministry said it would begin giving grants of $1,000 to $2,000 next week.