A spokeswoman for the defense minister, Yoav Gallant, said on Wednesday that another crossing into northern Gaza, near Zikim, a kibbutz, would open instead, and not the one near Erez. It was not clear if that was because of the damage at Erez.

Mr. Gallant told reporters that the government had approved a new border crossing and the use of the port of Ashdod, around 20 miles northeast of Gaza, for aid shipments, but he did not offer a time frame for either.

The United Nations says that a man-made famine is looming in Gaza, and many experts say that conditions in northern Gaza — which has mostly been cut off from aid deliveries since early in the war — already meet the criteria for a famine to be declared there. In that part of the territory, a few hundred thousand people are surviving on an average of 245 calories a day, according to Oxfam, an aid group.

Aid groups, the United Nations and a growing number of governments blame Israel for restricting aid into Gaza. U.N. figures show that an average of about 110 aid trucks have entered each day since Oct. 7. Though the daily average has risen since February, it is still far lower than the 500 trucks of commercial goods and aid that arrived in Gaza each day before the war.

Israel maintains that aid agencies have failed in their responsibility to distribute the aid. The groups say Israel has not created safe conditions that would allow them to distribute aid effectively.

The count of aid trucks Israel has allowed into Gaza recently has also been the subject of contention, raising questions about how to gauge the results of another pledge Israel made after the deadly airstrike against aid workers, which was to boost the number of trucks being screened at two existing crossings into southern Gaza.

Israel says the number has spiked, with COGAT writing in a social media post on Wednesday that an average of 400 trucks had entered per day over three recent days. The agency also posted photographs of street vendors selling cucumbers, potatoes and juice with the caption “market scenes in northern Gaza.”

People in northern Gaza have said in interviews that the little food available in street markets has long been out of reach for most, with many items priced at several times their original cost.

By contrast, U.N. data shows that a total of 533 aid trucks entered Gaza in the three days after Saturday. More broadly, U.N. figures show no increase in the daily average of trucks going into Gaza in the first week of April, compared to the previous week.

The reasons for the discrepancy are not clear, but one is the differing methods Israel and the United Nations use to track trucks, said Jens Laerke, a spokesman for the U.N. humanitarian office.

Trucks screened — and counted — by Israel at the two working border crossings usually enter Gaza only half full, after Israeli inspectors prohibit some of their contents, said Mr. Laerke. Once inside Gaza, they are unloaded, repacked as full trucks and sent to warehouses operated by the United Nations, which counts the number of full trucks that arrive, likely leading to a lower tally.

Other complications also mean that trucks often don’t pass through a crossing and arrive at a warehouse in the same day, meaning the daily counts at crossings and the warehouses often do not match, he said.

In a statement on Wednesday, COGAT criticized the U.N.’s “flawed counting method,” which it called “an attempt to conceal their logistical distribution difficulties.”

Previous Israeli promises to scale up aid have not greatly increased deliveries. Under U.S. pressure in mid-December, Israel reopened one crossing to Gaza, Kerem Shalom, for aid trucks, committing to permit 200 trucks a day to enter. But aid agencies say that stringent Israeli inspections have kept the numbers far lower than what is needed.

And Mr. Laerke and other aid officials said enormous challenges remain to distributing the aid inside Gaza, particularly to the north, where Israel has denied access for UNRWA, the main U.N. relief agency working in the territory.

Aaron Boxerman contributed reporting.