For many Gazans who have been forced to flee again and again, the situation on the ground may not change much. Israeli forces have repeatedly returned to conduct days-long operations in neighborhoods they already conquered during the initial offensive in an effort to crack down on renewed insurgencies by Palestinian militants.

The trigger for the evacuation orders and overnight Israeli attacks around Khan Younis appeared to be a barrage of roughly 20 rockets that the military said were fired from the area toward Israeli cities by Palestinian militants on Monday. Israeli forces struck back overnight after “enabling civilians to evacuate from the area,” the military said.

The United Nations estimated that roughly 250,000 people would have to flee a large swath of southern Gaza to comply with the Israeli military orders. Scott Anderson, a senior U.N. official, said the calculation was based on prewar population data and anecdotal observations on how many people had returned to the city.

For Gazans, recent operations intended to root out resurgent pockets of Hamas fighters are far from low-intensity. Hundreds of Palestinian fighters were killed in fighting in areas of northern Gaza such as Shajaiye, Jabaliya and Zeitoun, according to the Israeli military. In Jabaliya, over 60,000 people fled their homes, according to the United Nations, returning to find widespread devastation.

Israeli forces largely withdrew from Khan Younis in April after months of fighting as they were gearing up to invade Rafah farther south. In the relative calm that followed, many of the city’s residents went back home, some living in tents next to the rubble of their houses.

Suzan Abu Daqqa, 59, returned to her house on the southern outskirts of Khan Younis last month. It was relatively unscathed by the heavy Israeli bombardment that had destroyed large parts of the city, and it still had running water.

But on Monday evening, Ms. Abu Daqqa and her family heard that the Israeli military had yet again ordered the evacuation of the city’s eastern outskirts. The now-familiar sound of artillery fire began, she said, prompting her to flee northwest with relatives.

Thousands of people filled the streets of the demolished city on Monday night as they headed toward the Mawasi area near the coast, which Israel has designated as a “safer zone.”

“How long can we keep being ordered: Leave and come back, leave and come back?” said Ms. Abu Daqqa.

On Tuesday, Khan Younis residents said most of the explosions they could hear appeared to be farther south, in Rafah, indicating that at least for now, the fighting in their city was less intense. The wide-scale evacuation order, however, could potentially herald a renewed military operation there.

Amir Avivi, a retired Israeli brigadier general, said Israeli troops would seek to slowly whittle away at Hamas’s remaining fighters in the area, a process he said could take years. Over time, Israel hopes to erode Hamas’s forces so thoroughly that Gaza will take fewer and fewer forces to control, he said.

“Every time the terrorists manage to constitute themselves, there will be a raid to deal with them,” said General Avivi, who leads the hawkish Israel Defense and Security Forum. “These raids can last a few days or a week at a time — generally no more than a few days — and then you withdraw.”

General Avivi said for many Gazans, it would likely seem very similar to the current Israeli military campaign in the north.

“It won’t feel any different, save for perhaps in the forces applied and the number of troops,” he said.

U.S. officials have worked for months to prevent a war between Israel and Hezbollah, which is backed by Iran and has launched rocket attacks on northern Israel in solidarity with Hamas, the armed group that governed Gaza and started the current war when it attacked Israel on Oct. 7.

Fears of a full-scale, open war between Israel and Hezbollah have grown in recent weeks as exchanges of cross-border fire have intensified. Israeli officials have spoken publicly of shifting their military focus from Hamas to Hezbollah, a far more advanced and potent military threat.

Firas Maksad, a senior fellow at the Middle East Institute, wrote on X, there was still time for the key players to find a diplomatic solution. Mr. Hochstein ’s trip, he said, would likely happen on Wednesday. “The window for diplomacy is closing but not closed,” he said.

Mr. Blinken, speaking on Monday at the Brookings Institution, a nonpartisan think tank in Washington, D.C., that he did not believe that the key actors in the border conflict — Israel, Hezbollah and Iran — actually wanted to go to war, but that that’s what the “momentum” of the clashes could lead. U.S. officials fear that such a conflict could force the United States to come to Israel’s defense.

“No one actually wants a war,” Mr. Blinken said. He said that Iran, a determined foe of Israel, “wants to make sure that Hezbollah’s not destroyed and that it can hold onto Hezbollah as a card if it needs it, if it ever gets into a direct conflict with Israel.”

Some 60,000 Israelis have fled the area of the border clashes, many of whom have been living in Tel Aviv hotels for nine months. Referring to that situation, Mr. Blinken said that “Israel has effectively lost sovereignty in the northern quadrant of its country because people don’t feel safe to go to their homes.” The fighting has also displaced tens of thousands of people from southern Lebanon.

“Absent doing something about the insecurity, people won’t have the confidence to go back,” Mr. Blinken said. Resolving the issue, he added, will require an agreement to pull back forces from the border.

Mr. Blinken noted that Hezbollah has said that if a cease-fire were reached in Gaza, it would stop firing into Israel. That “underscores why a cease-fire in Gaza is so critical,” he said. But the latest round of negotiations between Israel and Hamas appear deadlocked.

Mr. Hochstein has met in recent weeks with Israeli officials and also with Lebanese officials, who can pass messages to and from Hezbollah, in an effort to negotiate a Hezbollah pullback to a position far enough from the border to satisfy Israel. In return, Israel might withdraw from some disputed border areas, and the U.S. could provide economic assistance for southern Lebanon, analysts say.