U.N. officials warned on Sunday that pressure was mounting near Gaza’s border with Egypt, an area where tens of thousands of Palestinians have tried to flee Israel’s military campaign.

Gazans have flocked to the area seeking safety and supplies. Thousands continue to arrive, heeding evacuation orders from Israel’s military that named parts of Rafah, the region next to the crossing, as a place to go for refuge, even as Israel expands its ground operation in the south.

But relief is in short supply. Aid officials have warned of “extreme” overcrowding and a “dire” situation in the border area, and have raised alarms about the spread of disease. Deteriorating conditions could push Palestinians over the border into Egypt, two U.N. officials warned on Sunday.

“The health care system is collapsing,” António Guterres, the U.N. secretary general, warned on Sunday in Qatar, saying that there was “no effective protection of civilians in Gaza.”

“I expect public order to completely break down soon, and an even worse situation could unfold, including epidemic diseases and increased pressure for mass displacement into Egypt,” he added.

Philippe Lazzarini, the commissioner general of the United Nations Relief and Works Agency, which takes care of Palestinian refugees, said that even if there was no “deliberate policy,” Israel’s military operations were “putting more and more pressure for this type of scenario to unfold.”

“The population has been pushed more and more into tinier and tinier and tinier pieces of land in the Gaza Strip, and there is no way that this piece of land will be able to accommodate such a high number of people,” he said in an interview on Sunday.

Gazans who have fled to the Rafah area thinking that the southern part of the Gaza Strip would be safe have found airstrikes there, too.

“We have no other place — and no safe place,” said Ziad Obeid, a senior civil servant for the Palestinian Authority, who said he had come to the outskirts of Rafah with his family after being displaced farther and farther south since the beginning of the war.

“We are fighting day and night just to get some bread, water and some vegetables,” he said, adding that he had struggled without success to find a few eggs for his elderly mother.

The Israeli government has not publicly called for large numbers of Gazans to move to Egypt. But diplomats have said that, in private, Israel has pushed for them to be housed in Egypt for the duration of the war — augmenting Palestinian fears of a permanent expulsion.

Egypt is adamantly opposed to the idea, partly out of concern that the country — already facing a precarious economic situation — could be destabilized by an influx of refugees.

Egypt and other Arab governments also worry that such a move could give Israel a pretext to carry out a forced displacement of Gazans that could irreparably damage the struggle for Palestinian statehood. Gazans must “stay steadfast and remain on their land,” President Abdel Fattah el-Sisi of Egypt said in October.

In moves that appeared to reflect concerns that the border could be breached, Egypt’s army has erected sand barriers and stationed tanks and other military vehicles along the border at Rafah, according to two people who live near the Egyptian side of the border. They spoke on the condition of anonymity because of the sensitivity of the issue.

In 2008, tens of thousands of Palestinians crossed into northern Egypt after Hamas broke down parts of the border fence — forcing a temporary end to the blockade of Gaza that had followed Hamas’s then-recent takeover of the territory.

Patrick Kingsley and Nada Hussein contributed reporting.