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Nearing the end of a whirlwind Middle East trip this week, Secretary of State Antony J. Blinken finished meetings<\/a> with the Israeli president and relatives of American hostages held by Hamas, left his beachside hotel in Tel Aviv and shook hands with protesters gathered outside.<\/p>\n He looked them in the eye and said there was a new hostages-for-cease-fire deal<\/a> on the table that Hamas should take.<\/p>\n \u201cBringing your loved ones home is at the heart of everything we\u2019re trying to do, and we will not rest until everyone \u2014 man, woman, soldier, civilian, young, old \u2014 is back home,\u201d he said.<\/p>\n That public show of empathy with frustrated protesters is something that Prime Minister Benjamin Netanyahu has avoided since the war began in October. And, lately, he has focused his recent public comments on an imminent ground offensive \u2014 an invasion of the city of Rafah in southern Gaza \u201cwith or without\u201d a cease-fire deal<\/a>, as the Israeli leader put it on Tuesday.<\/p>\n<\/div>\n<\/div>\n Though it was not the first time Mr. Netanyahu has promised to invade the last Hamas stronghold in Gaza, U.S. officials were taken aback by the timing of the comment. Threatening an offensive in Rafah can put pressure on Hamas to take the deal \u2014 but only if Hamas leaders think freeing hostages for Palestinian prisoners and a six-week pause in fighting could eventually lead to a permanent cease-fire and avert a bloody battle in Rafah, where more than a million displaced Gazans have sought shelter, the officials say.<\/p>\n<\/div>\n<\/div>\n Nearly seven months into the war, the stated aims and diplomatic efforts of the United States and Israel seem further apart<\/a> than ever \u2014 a gap that continues to widen under the domestic political imperatives of President Biden and Mr. Netanyahu.<\/p>\n Mr. Biden and his top aides envision a path that involves Hamas freeing about three dozen hostages within weeks; the two sides enacting a temporary cease-fire that leads to a permanent one and more hostage releases; and prominent Arab nations, including Saudi Arabia, agreeing to take part in reconstruction and security efforts, as well as in normalizing diplomatic relations with Israel.<\/p>\n Israeli officials have shown some flexibility recently on the terms of the cease-fire deal, saying they would reduce the number of hostages Hamas would have to release in the initial round to 33 from 40.<\/p>\n<\/div>\n<\/div>\n Yet, even as Israel yielded on these points, Mr. Netanyahu has rejected the idea of a permanent cease-fire and doubled down on his public vow to eradicate Hamas and many fighters that he says remain in Rafah \u2014 despite a widespread belief among U.S. officials that his goal is unattainable.<\/p>\n U.S. officials oppose invading Rafah and say Israel should carry out precise operations against Hamas leaders, not a major assault. When Mr. Blinken met with Mr. Netanyahu in Jerusalem on Wednesday, he reiterated the \u201cclear position\u201d of the United States on Rafah, said Matthew Miller, a State Department spokesman.<\/p>\n<\/div>\n<\/div>\n The pressures on the Biden administration are also clear. Mr. Biden\u2019s liberal voting coalition could fracture as opposition builds to his unwavering support of Israel in the war, jeopardizing his chances of defeating Donald J. Trump, the Republican contender, in November. The students protesting Mr. Biden\u2019s policy on American college campuses and resultant police crackdowns have further thrust the issue into the spotlight.<\/p>\n And the United States finds itself deflecting criticism from Arab partners and governments across Asia, Africa and Latin America, and shielding Israel from pro-Palestinian resolutions in the United Nations. Amid cries of hypocrisy against Washington, it is clear that Mr. Biden\u2019s backing of Israel will make it harder for him to win support for American policies aimed at countering Russia and China<\/a>, particularly in the nations of the global south.<\/p>\n<\/div>\n<\/div>\n Mr. Blinken is grappling with the challenges. On Monday, the first day of his current Middle Eastern tour<\/a>, in meetings with Arab and European officials in the Saudi capital, Riyadh, he steered discussions toward a hostage release and plans for postwar reconstruction in Gaza. He made humanitarian aid the theme of his stop in Jordan the next day.<\/p>\n When reporters asked him about Mr. Netanyahu\u2019s insistence on a Rafah offensive, Mr. Blinken said the cease-fire deal and humanitarian aid were the \u201cfocus\u201d of American efforts.<\/p>\n The Israeli protesters outside Mr. Blinken\u2019s hotel in Tel Aviv were on the same wavelength. They have placed their hopes in the American government rather than their own to end the crisis, which began when about 1,200 Israelis were killed in the Hamas-led attacks on Oct. 7 and about 250 were taken hostage. More than 34,000 Palestinians have been killed in the Israeli military\u2019s retaliatory air campaign and ground invasion.<\/p>\n \u201cS.O.S. U.S.A., only you can save the day,\u201d the protesters chanted. \u201cThank you, Biden, thank you, Blinken.\u201d<\/p>\n<\/div>\n<\/div>\n Mr. Biden and Mr. Netanyahu are also clashing over what the Americans call a long-term political solution to the decades-old Israeli-Palestinian conflict. The Americans are working on a plan to have Saudi Arabia and perhaps other Arab nations agree to normalize relations with Israel \u2014 but only if the Israeli government commits to a concrete path with firm deadlines to the founding of a Palestinian nation. Mr. Netanyahu opposes that, as do many Israelis.<\/p>\n<\/div>\n<\/div>\n Still, Mr. Biden maintains his general support for Israel in the war, and he has not placed conditions on military aid or weapons sales<\/a>, something that even centrist foreign-policy analysts<\/a> and former officials in the United States are calling for.<\/p>\n Mr. Netanyahu, who is clinging to power despite a slump in his international and domestic standing, faces a range of seemingly mutually exclusive choices. He is caught between the competing pressures applied by the Biden administration and the far-right members of his governing coalition, whose support is crucial for the survival of his government.<\/p>\n His hard-right ministers are threatening to quit if the long-touted Rafah operation is suspended. Bezalel Smotrich, the ultranationalist finance minister, has described<\/a> the hostage deal on the table as \u201ca dangerous Israeli capitulation and a terrible victory for Hamas.\u201d Itamar Ben-Gvir, the far-right national security minister, said on Tuesday that he had \u201cwarned\u201d Mr. Netanyahu of the consequences of not going into Rafah and instead agreeing to a \u201creckless deal\u201d ending the war.<\/p>\n Centrists who joined Mr. Netanyahu\u2019s government in October, bringing broader popular legitimacy to its war efforts, have given notice that they will not tolerate decisions based on political considerations rather than the national interest.<\/p>\n<\/div>\n<\/div>\n The Israeli public simultaneously yearns for the return of the hostages and craves the defeat of Hamas, even as it is split over the prospects of an absolute victory.<\/p>\n