CNN — Each day the pause in Israel’s war with Hamas is extended saves lives.

A second extension of the truce, lasting one day, came into force early Thursday. But the lull in fighting also sharpens the moral, political and military dilemmas that will play out in the almost inevitable return to full-scale hostilities – including some apparent strategic and humanitarian differences of emphasis between the Biden administration and the Israeli government.

While the truce has so far been surprisingly successful – given that it is taking place with both Israel and Hamas seeking the other’s elimination – there is an unmistakable sense that a fateful moment is approaching within days when Israel will decide on how long it can hold off its scorching military offensive.

That means the debate about what happens next in Gaza is increasingly urgent, even as the US seeks to extend the pause in fighting in the medium term and Israel seeks to temper expectations of restraint in the days to come.

These profound questions are also taking place against a backdrop of deepening tragedy in a cruel war, even if the emergence of hostages has offered fleeting moments of joy amid the horror. In one wrenching development on Wednesday, Israel said it was looking into a claim that the youngest Israeli hostage, 10-month-old Kfir Bibas, his brother and his mother are no longer alive. Uncertainty, meanwhile, surrounds the future of the remaining hostages, mostly male, that Israel believes remain in Gaza. The humanitarian crisis in the territory is worsening with the World Health Organization warning that more Gazans could die from disease than bombing if an already rudimentary health service is not urgently repaired. And as growing unrest rocks the occupied West Bank, the Israeli military killed two Palestinian children when it opened fire in the city of Jenin, according to the Palestinian Ministry of Health – the latest of more than 240 Palestinians that the ministry says have been killed by Israeli soldiers and settlers in the West Bank since October 7.

Will Israel listen to US pleas for a more surgical approach?

Israel has made no secret of its aim since the October 7 Hamas terror attacks that killed 1,200 people — the irrevocable eradication of the Islamist group designated by the US as a terror organization, which controls the Gaza Strip. Prime Minister Benjamin Netanyahu’s government viewed the Hamas attacks as an existential threat to Israel and the Jewish people and argues it has no choice but to crush Hamas completely.

The Israeli leader on Wednesday dismissed the idea that the prolonged pause in fighting would make it more difficult strategically and morally for Israel to resume its relentless action against Hamas. “Over the past few days I’ve been hearing this question – will Israel go back to fighting after maximizing this phase of returning our hostages? So my answer is unequivocal – Yes,” he said.

But Israel’s initial assault on Hamas led to huge civilian carnage in the densely populated Palestinian enclave in the first phase of the war, sending tens of thousands of protesters into the streets in the US and across the world while heaping political pressure on President Joe Biden from inside his own electoral coalition.

The likely prospect that a second wave Israeli offensive against Hamas strongholds in southern Gaza would be even more bloody now threatens to open gaps between Washington and Netanyahu’s government and military leaders. CNN’s MJ Lee, Jennifer Hansler and Katie Bo Lillis reported Wednesday that US officials, including Biden, told the Israelis they don’t want to see a repeat of air strikes that led to massive destruction and terrible scenes of civilian casualties. Israel must be more “cautious, more careful, more deliberate and more precise in their targeting,” one senior administration official said.

In the days after the October 7 attacks, Biden hugged Israel and Netanyahu close, traveling to the Jewish state to mourn with the victims of the horror. Will the Israeli prime minister pay any more attention to Biden’s entreaty to do more to protect Palestinian civilians than the passing consideration that he gave it in the first days of the conflict?

This is an issue likely to lead to intense discussions on Thursday between US Secretary of State Antony Blinken, who’s in Israel, and top Israeli officials.

Israel’s moral and military dilemmas

The Israeli government is being pulled in two directions that may be irreconcilable – the desire to get all of the hostages back and the incentive to press on with its military operation after a pause that offered Hamas a chance to regroup and prepare for a new assault.

At home, the Israeli prime minister, beset by deep unpopularity after the surprise Hamas attacks, is also being pulled between growing political pressures from hostage families, who want their loved ones released, and his right-wing coalition members, who are advocating for harsh action amid frustration that the pause has allowed Hamas to use hostages to regain control of the tempo of the crisis. And Netanyahu also faces the growing possibility of a clash between his desire to target Hamas and US anxiety over another round of huge civilian casualties in Gaza. American support would be even more crucial for Israel in a second phase of fighting because foreign powers are likely to strongly criticize the Netanyahu government if it is seen to re-ignite hostilities.

Israel’s heavy-handed military tactics are also coming under scrutiny amid fears that more civilian casualties will seed a new generation of fury against the Jewish state that will eventually translate into recruitment for extremist groups and terrorism.

Retired US Army Lt. Col. Daniel Davis noted that the Israeli leader sees the elimination of Hamas as the key to bringing peace to Israel but warned, “What he is doing with the military will not bring that peace.” He told Kasie Hunt on CNN International’s “State of the Race” that “I’ve seen it myself in Afghanistan several times, to where the more Taliban you kill, the more the enemy, the more you make, especially when you’re killing so many people.”

Israel says that it takes pains to avoid killing civilians. Hamas has embedded its forces within the civilian population, using infrastructure like hospitals and apartment blocks as cover. Senior Israeli figures argue that while Washington wants to see more forensic targeting of Hamas if the battle resumes, such an approach is not always feasible given conditions in Gaza.

“We’re not magicians,” former Israeli Prime Minister Naftali Bennett said Tuesday, accusing Hamas of deliberately sacrificing Palestinian civilians to stir global anger against Israel. “If there was some magical solution where we could tweeze our people out and just hit the rocket launcher that’s shooting rockets at Israelis, we would do it,” Bennett told CNN’s Jake Tapper. “We do try to reduce unnecessary civilian casualties, but the reality is that there’s no magic.”

One possible approach that US and Israeli officials are deliberating is for Israel to allow civilians it sent to southern Gaza early in the war to head back north, one senior US official told CNN. Multiple officials have also raised the need for the creation of safe zones for civilians in the south, CNN reported. But there are practical difficulties with such a plan. For one thing, swathes of northern Gaza have been devastated by the Israeli advance, as new drone footage of mile upon mile of buildings turned into rubble shows. In an enclave surrounded by Israel and with the border to Egypt closed, there is nowhere for millions of people to go.

While a debate is boiling over the tactics to be used in a second phase of the war, the very idea of starting the fighting again while hostages are still being held by Hamas is horrifying family members of those still held. The pause in the fighting over the last week has offered both Netanyahu and Biden some political relief on this issue but that would end as soon as the guns begin firing.

Yehuda Beinin – the father of Liat Beinin, a dual US-Israeli national freed by Hamas on Wednesday – voiced a growing concern among hostage families that the remaining hostages are not the first priority of Netanyahu’s government. “This naturally would create a great deal of fear that the hostages will again come under some kind of danger, as a result of renewed Israeli bombing,” said Beinin, whose son-in-law Aviv Atzili, is still believed to be a hostage.

Biden administration steps up diplomacy

The Biden administration’s current priorities are:

—  A further extension of the truce.

—  The release of all the hostages from Gaza.

—  The alleviation of a horrendous humanitarian crisis in the enclave.

—  Supporting Israel’s effort to prevent future terror attacks.

—  Turning the focus toward post-war governance of Gaza ahead of an effort to tackle the long-dormant question of diplomacy to halt the Israel-Palestinian conflict.

Potential difficulties with Israel lie in the fact that such goals – or their urgency – may not always intersect with those of the Netanyahu government. Biden has a huge stake in the conflict for humanitarian, political and geopolitical reasons. The president noted in a statement on Wednesday that the US, along with Qatar, had been instrumental in negotiating the pause that had freed scores of hostages, and delivered substantial humanitarian aid to help “the innocent people of Gaza.”

Biden’s support for Netanyahu has cost him politically at home and abroad. US foreign policy goals in the Arab world and elsewhere risk being compromised as governments react to outrage at the civilian death toll in Gaza. Core initiatives like the bid to cement peace between Israel and Arab states have been seriously damaged. Domestically, scenes of Palestinian deaths have split the Democratic Party and raised fears over whether younger and more progressive voters already cool toward Biden will show up in the numbers he needs in November 2024. Less of a concern, but still significant, are attacks from Republicans at the first sign he is trying to constrain Netanyahu.

While there is no sense that unshakable US support for Israel is at risk, the real danger of growing differences between the two governments about the future conduct of the war could introduce new tensions into the relationship.

The vital national interests of the United States and Israel are not always or irrevocably aligned. So whether Netanyahu has political room to maneuver on military strategy or the inclination to ease some of the pressure on Biden will be closely watched in the coming days.

How far will Israel go in testing Biden’s loyalty, forbearance and political viability if the war heats up again? And if push comes to shove, will Netanyahu’s desire to eliminate Hamas override all other considerations?