The Israeli military has limited time to carry out its operations in Gaza before anger among Arabs in the region and frustration in the United States and other countries over the spiraling civilian death toll constrain Israel’s goal of eradicating Hamas, U.S. officials said this week.

As senior Biden administration officials push Israel to do more to minimize civilian casualties, Gen. Charles Q. Brown Jr., the chairman of the Joint Chiefs of Staff, said on Wednesday that he was worried each civilian killed in Gaza could generate future members of Hamas.

General Brown did not call for a cease-fire. But when asked by reporters traveling with him to Tokyo if he was worried that high civilian casualty numbers would generate future Hamas militants, he replied, “Yes, very much so.”

His comment offered a rare glimpse of divisions between Israel and the Biden administration, which has declared its support for Israel’s military campaign even as the civilian death toll has increased. It came as the United Nations secretary general, António Guterres, said that the number of civilians killed in the Gaza Strip showed that there was something “clearly wrong” with Israel’s military operations against Hamas.

Israel launched a ground invasion after Hamas and Palestinian Islamic Jihad fighters rampaged through southern Israel on Oct. 7, indiscriminately killing women, children, babies and the elderly. More than 1,400 people were slain and more than 240 were taken hostage and ferried to Gaza. On Wednesday, Israeli investigators said that “victims were tortured, physically abused, raped, burned alive and dismembered.”

The level of carnage has deeply shaken Israel and shaped its military response. The country’s leaders have vowed to eliminate Hamas, which is committed to the destruction of Israel, and to kill everyone implicated in the Oct. 7 atrocities.

But the longer the Israeli military campaign continues, the greater the chance that the conflict will spark a wider war, several officials in the Biden administration said. In addition, several officials said that Israel’s forceful response to the Hamas attacks has fueled sympathy around the world for the Palestinian cause even as Israel continues to bury its dead.

Time and terrain have worked against Israel in Gaza. Hamas has used civilians as human shields and positioned underground bunkers, weapon depots and rocket launchers under or near schools, mosques and hospitals.

All of this has raised the risk of civilian casualties when the Israeli army targets Hamas sites. In late October, Israeli fighter jets targeted a tunnel network under Jabaliya, a densely populated neighborhood, killing a central figure in Hamas’s Oct. 7 terrorist attacks. But local officials say dozens of civilians were also killed and hundreds were wounded in the strike.

Israel’s rapid decision to launch ground operations in the tightly packed enclave left little time for extensive advance planning to mitigate risks to civilians and all but guaranteed a high civilian death toll, U.S. officials said. In the first month, some 10,000 people — around 40 percent of them children and teens — have been killed, according to the health ministry in Gaza, which is governed by Hamas.

The longer the bombing campaign continues, the more isolated Israel — and its ally, the United States — could become as countries around the world call for a cease-fire. Prime Minister Benjamin Netanyahu of Israel has said that there will be no cease-fire until all the hostages are freed. But he also seemed to understand that Israel did not have unlimited time to achieve its objectives.

On Tuesday, Mr. Netanyahu said in a statement that Israel was working diplomatic channels to “provide the I.D.F. with international maneuvering room for continued military activity.”

The issue is coming to a boiling point as Israel Defense Forces close in on Al Shifa Hospital in the center of Gaza City. Israeli and American officials say that senior Hamas operatives have placed a command and control center underneath the hospital.

I.D.F. officials say they do not intend to back off.

The Israeli defense minister, Yoav Gallant, said on Tuesday that the Israeli military was “tightening the noose around Gaza City” and that Yahya Sinwar, the Hamas leader in Gaza, was “hiding in his bunker.”

An American intelligence official said Mr. Sinwar’s exact location was unclear and it was unknown what Israeli security and intelligence officials had gleaned about his whereabouts. But several other national security officials said the hospital is a hugely challenging target for the I.D.F. Striking it, one senior American official said, could kill hundreds or even thousands of Palestinians.

Israel’s strikes and encirclement of Gaza City have prompted thousands of Palestinian civilians in the northern Gaza Strip to begin moving southward. On Tuesday, the I.D.F. opened a humanitarian corridor for a few hours, and thousands of Palestinians, many waving white flags, took advantage.

The Israeli military now says that it will guarantee safe passage during a daily window, the White House announced on Thursday, after President Biden had pressed for humanitarian pauses.

Gen. Joseph L. Votel, the former leader of the U.S. military’s Central Command, said the United States had established corridors for people to flee Raqqa, Syria, in 2017 during the fight against the Islamic State. He said that Central Command officials had planned the Raqqa campaign for months before American-backed forces went in, partly to try to limit civilian casualties.

The lead-up to the fight to retake the Iraqi city of Mosul from the Islamic State in 2016, another official said, took nine months, partly for the same reason.

Senior American military leaders, in near-daily phone calls, are trying to push their Israeli counterparts to be “more calculating and precise” in targeting, one official said. Other officials have urged Israel to use 250-pound satellite-guided bombs instead of 1,000- to 2,000-pound munitions on military targets in Gaza.

Current and former U.S. military commanders said the Israeli military’s encirclement of Gaza City, effectively splitting the Gaza Strip in half, would make it harder for Hamas to control the enclave.

Israel’s decision two weeks ago to hold off on a full-scale invasion of Gaza and instead carry out a more deliberate, phased ground offensive aligned with suggestions from Defense Secretary Lloyd J. Austin III to his Israeli counterparts, officials said.

But Israel has also dropped thousands of tons of bombs in its punishing air campaign while protecting its ground troops from Hamas ambushes. About three dozen Israeli soldiers have died, an indication that the Israeli army is moving cautiously on the ground while warplanes and artillery pound targets.

Encircling Hamas in one of its strongholds and cutting off its resupply lines and communications forces the group to use up existing supplies and exhausts its fighters, said former U.S. commanders who have fought in urban battles in Iraq and Syria.

Gen. Kenneth F. McKenzie Jr., a retired head of U.S. Central Command, said the Israeli military set five major objectives for the Gaza campaign: dismantle Hamas, minimize civilian casualties, reduce the risks to its own troops, recover the hostages and avoid widening the war beyond Gaza.

Israel is achieving most of those goals, General McKenzie said, carrying out a focused campaign to cut off Gaza City from the rest of the enclave, suppressing Hamas rocket fire into Israel and, with U.S. assistance, searching for the hostages, many of whom are believed to be held in a vast tunnel complex — all the while trying to minimize civilian casualties.

“The Israeli campaign has been very deliberate,” General McKenzie said.

Strategically, however, he said that time “is not necessarily on Israel’s side.” Criticism of Israel is growing as memories of Hamas’s Oct. 7 attacks fade and images of Gaza in ruins and of civilian casualties dominate news feeds. That is putting pressure on the I.D.F. to inflict damage on Hamas as quickly as possible.

General Brown said the longer the war goes on, the more challenging it will become for Israel.

“Every conflict that I’ve been involved with throughout my military career, with the exception of probably Desert Shield/Desert Storm, has gone a lot longer than most people would have imagined,” he said, referring to the 1991 Persian Gulf war.

Israeli commanders say they are taking into consideration the safety of Palestinian civilians and the hostages as they carry out the campaign. But some current and former U.S. military officials questioned whether Israel was prioritizing the safety of civilians.

Or as Christopher Costa, a former Army intelligence officer and White House counterterrorism official, put it, “The Israeli risk tolerance — calculus — seems to be exceedingly high in terms of the criticism they’re willing to accept right now for civilian casualties while operating in a complex urban and densely populated environment.”

In a phone call on Oct. 30 — one day before Israel bombed the Jabaliya refugee camp — General Brown emphasized to his Israeli counterpart, Lt. Gen. Herzi Halevi, the importance of mitigating risks to civilians when bombing suspected Hamas leaders hiding in underground tunnels, given the likelihood that the area above ground will collapse when a tunnel is struck, said one official briefed on the conversation.

Yaakov Peri, a former head of Shin Bet, the Israeli internal security service, said the military and intelligence agencies would kill the Hamas commanders who carried out the Oct. 7 attacks. But like General Brown, he worries that Israel is creating a new generation of fighters.

“We’ll be fighting their sons in four or five years,” Mr. Peri said.