Out of Gaza's 640 schools, 18 were flattened and 280 suffered some damage during Israel's three-week offensive against Hamas seven months ago, the groups said in a statement.
Since the operation, Israel has refused to allow construction materials into Gaza, arguing that Hamas could divert iron rods and concrete to build rockets and bunkers.
Both Israel and Egypt have kept Gaza's borders largely closed since Hamas seized control of the territory by force more than two years ago.
The border blockade and the operation have further burdened Gaza's education system, which even before the Hamas takeover suffered from serious overcrowding. Many schools have been running morning and afternoon shifts for lack of space.
About 500,000 of Gaza's 1.4 million residents are of school age. Of those, nearly half attend schools run by the UN Relief and Works Agency, which cares for Palestinian refugees and their descendants.
About 6,000 students will have to be reassigned to different schools because their old ones were rendered unusable by the war, said Numan Sherif, an official in the Hamas Education Ministry.
"The problem is the blockade," Sherif said. "There's money to rebuild, but we don't have access to basic materials, or even furniture. We can't fix toilets or the wiring in schools."
He said Gaza would also need 100 new schools just to keep pace with population growth.
A UN official, Marixie Mercado, said aid groups meet regularly with Israeli defense officials on the issue. Mercado said defense officials allowed in text books, paper and some teaching kits.
Foreign Ministry spokesman Yigal Palmor said the aid groups could not guarantee that construction materials intended for schools would not be diverted by Hamas operatives.
"The point is, what will they do with iron, what will they do with cement?" asked Palmor. "Will it go to the schools? We have a good reason to believe it won't. This is not an abstract fear," he said.
John Ging, the top UN aid official in Gaza, challenged that argument, noting that UNRWA keeps track of the supplies allowed into Gaza by Israel.
"We account for every sack of flour and we can equally account for every bag of cement," he said. "It's just a matter of political will to move forward on this issue. We'd like to get on with the job, and then be held accountable on whether we are achieving it or not".