After the Israeli military issued its evacuation order, Prime Minister Rishi Sunak of Britain reiterated that he remained “deeply concerned” about an invasion, while Saudi Arabia’s foreign ministry warned Israel against advancing what it called a “bloody and systematic” campaign to storm all of Gaza and displace its residents.

In Rafah, some Palestinians quickly dismantled their tents in the pouring rain and began leaving. Others questioned whether it was safe to leave. Gazans and aid groups have said that the Israeli military has bombed areas that it has previously designated as safe for civilians.

Nidal Kuhail, 29, a former resident of Gaza City, said he was overcome with anxiety and divided over what to do. The tent he was sheltering in was not in the area of Rafah covered by Israel’s evacuation order.

“If we have to leave, we will be entering the unknown,” Mr. Kuhail said. “Are we going to have a place to go? Are we going to be able to find a place to set up the tent?”

Workers for UNRWA, the main United Nations agency that aids Palestinians in Gaza, estimated on Monday that about 200 people an hour were fleeing the evacuation zone, said Sam Rose, the agency’s director of planning.

Israel was telling Palestinians to move to an area that includes al-Mawasi, a coastal section of Gaza it has advised people to go to for months, as well as areas farther north along the coast to Deir al-Balah. The military said the area had field hospitals, tents and larger supplies of food, water and medicine.

Israel was not calling for a “wide-scale evacuation of Rafah,” a military spokesman, Lt. Col. Nadav Shoshani, said on Monday. “This is a very specific scoped operation at the moment to move people out of harm’s way.”

But Mr. Rose said that the area would not be able to safely accommodate all the civilians who have sheltered in Rafah, in part because parts of it are littered with unexploded bombs.

Going there would also move them farther from the entry points for desperately needed food, water, medicines and other supplies, which aid agencies have struggled to distribute around Gaza.

“They would basically be going back to oblivion,” Mr. Rose said.

Mahmoud Mohammed al-Burdeiny, 26, said he thought Israel had been using the threat of a Rafah invasion as a bluff to get a better deal from Hamas in cease-fire talks. But now the danger felt real, he said.

So Mr. al-Burdeiny and his wife began to pack their belongings and plan for the worst. They could take the doors of their house with them to use as shelter, they realized. And they could take apart their furniture to use as firewood.