The Biden administration has been leading the latest push for a cease-fire deal between Israel and Hamas, with Egypt and Qatar acting as mediators, and has presented a “bridging proposal” meant to close, or at least narrow, the gaps between the sides.

But a visit to the region this week by Secretary of State Antony J. Blinken ended without any apparent resolution to major sticking points — and in acrimony, with Mr. Netanyahu and Hamas officials trading blame for obstructing progress.

One main focus of disagreement is Mr. Netanyahu’s insistence on maintaining an Israeli military presence along the Philadelphi Corridor, a narrow piece of land along Gaza’s southern border with Egypt. The Israeli leader has argued that the area has served as a main conduit for weapons smuggling into Gaza, and that abandoning it would allow Hamas to quickly rearm.

Egypt and Hamas strongly object to continued Israeli control of the area, calling for a full Israeli withdrawal from Gaza, and Israeli security officials have suggested that other solutions can be found.

Israeli negotiators were taken aback and, in some cases, angered by Mr. Blinken’s statements this week that Mr. Netanyahu had accepted the American bridging proposal and that it was now up to Hamas, according to people familiar with the talks, who spoke on the condition of anonymity to share the thinking within Israel’s top security and decision-making circles.

Those statements — which came after a nearly three-hour meeting on Monday between Mr. Blinken and Mr. Netanyahu — made it appear that the Israeli prime minister and the Biden administration saw eye-to-eye on an issue that Hamas clearly would not accede to, one of the people said, dooming the prospects of an agreement. Instead, they said, Mr. Blinken could have just called more vaguely for flexibility on both sides.

The details of the bridging proposal have not been made public. An Israeli official with knowledge of the talks, who was not authorized to speak about them publicly, cautioned that few people really know what is being said behind closed doors, and said that talks were continuing at various levels as part of a difficult negotiation.

Before departing Qatar, the last stop on his regional tour, Mr. Blinken apparently engaged in some damage control, saying, “The agreement is very clear on the schedule and the locations of I.D.F. withdrawals from Gaza,” referring to the Israeli military. He added: “Israel has agreed to that.”

In a follow-up call on Wednesday with the emir of Qatar, Sheikh Tamim bin Hamad Al-Thani, Mr. Blinken underscored that the bridging proposal addressed the remaining gaps “in a manner that allows for swift implementation of the deal,” according to a State Department readout of the call.

Mr. Blinken and the emir “welcomed the next round of talks in Cairo,” the statement added, without specifying a date. A meeting between Mr. Blinken and the emir planned for earlier in the week in Qatar was called off because Mr. Al-Thani was feeling unwell.

The Biden administration, Egypt and Qatar have also pushed for a cease-fire in hopes that it will calm tensions between Israel and Iran, which has pledged to retaliate for the killings last month of senior leaders of Hamas and Hezbollah, two of its proxy forces in the region. Mohammed bin Abdulrahman Al-Thani, the Qatari prime minister, will travel to Iran in the coming days for talks with senior officials, Iran’s semiofficial Tasnim news agency reported on Thursday.

Rawan Sheikh Ahmad contributed reporting from Jerusalem.

The tanker, the Sounion, was sailing about 90 miles west of the Yemeni port of Hudaydah early on Wednesday when two small boats approached it, said a statement on social media from Britain’s maritime trade agency, which is based in Dubai.

“The first craft had three to five persons onboard while the second had approximately 10,” the statement said. “The two small craft hailed the merchant vessel, leading to a brief exchange of small-arms fire.”

The small craft retreated and the ship was then hit by three “unidentified projectiles,” starting a fire and causing the ship to lose engine power, it said. It was not clear on Thursday whether the fire had been extinguished.

The crew were rescued by a ship from an E.U. military mission, Operation Aspides, that has been mounted in response to attacks by the Houthi militia, according to a statement the mission posted on social media Thursday.

The statement included a photograph showing an enclosed lifeboat, typical in the oil industry, bobbing in the water and a second photo of crew members aboard a rescue speedboat.

After coming under attack, the ship’s captain called for help. The E.U. military destroyed what it described as an “unmanned surface vessel” that had posed an imminent danger to the Sounion. The crew was being transported to Djibouti, the statement said.

The stricken vessel, which was carrying 150,000 metric tons of crude oil, was still floating and had become a “navigational and environmental hazard,” according to the E.U. statement. A statement from Delta Tankers, the ship’s operator, said it would be moved to a safer place for repairs. It was not clear whether any oil was leaking.

The Houthi militia, which is backed by Iran, began firing late last year on ships entering the Red Sea en route to the Suez Canal, which is a vital artery for vessels moving between Asia, Europe and the eastern United States.

The United States and Britain have launched strikes against the Houthis in response, but analysts have said they have done little to damage the militia’s military infrastructure. The attacks have continued, forcing ships to find alternative routes and disrupting global trade. Oil prices on Thursday were little changed, trading near their lowest levels of the year.

Over the course of dozens of attacks, at least two vessels have sunk and at least three crew members have been killed.

Jason Karaian contributed reporting.

Hours later, Rachel Goldberg and Jon Polin, Chicago natives, took to the convention stage. Wearing pieces of masking tape with the number 320 — the number of days their son has been a hostage — they were greeted with a standing ovation and chants of “Bring them home!” Ms. Goldberg briefly broke down in tears.

“At this moment, 109 treasured human beings are being held hostage by Hamas in Gaza,” she said. “They are Christians, Jews, Muslims, Hindus and Buddhists.”

Ms. Goldberg noted that the hostages were from 23 countries, including eight Americans, one of which is her only son, who like the Democratic nominee, Vice President Kamala Harris, was born in Oakland, Calif. Part of his left arm was blown off by a Hamas grenade on the day he was abducted.

“This is a political convention,” Mr. Polin said. “But needing our only son and all of the cherished hostages home is not a political issue.” He called for a cease-fire, saying, “There is a surplus of agony on all sides of the tragic conflict in the Middle East. In a competition of pain, there are no winners.”

Keith Ellison, attorney general of Minnesota — and in 2006, the first Muslim elected to Congress — also called for a cease-fire in Gaza and for the return of the hostages.

Uncommitted’s campaign to make the war in Gaza a focus of the convention comes as the death toll there has surpassed 40,000, according to Gazan health authorities. Hopes for the survival or recovery of the more than 100 hostages, some believed to be dead, who are still being held in the enclave have grown dimmer. The Israeli military announced on Tuesday that its forces had recovered the bodies of six hostages from a tunnel in Gaza, and the latest effort by the Biden administration to negotiate a cease-fire between Israel and Hamas appeared to have reached a stumbling block, with many issues unresolved.

“The D.N.C. coincides with a critical moment in the negotiations,” Johnathan Dekel-Chen, the father of Sagui Dekel-Chen, a 35-year-old Israeli American hostage, said in an interview. He has been at the convention with the relatives of other American hostages talking to Democratic leaders about the need for a deal, arguing that without an agreement in the coming weeks, there’s a high risk of an escalation of conflicts in the Middle East.

“The moment is critical not just for the hostages but for the region,” he said.

Margaret DeReus, who heads the Middle East Institute for Understanding’s policy project, a nonprofit advocacy group working with Uncommitted, said in a phone interview from Chicago that as she and her colleagues have walked the convention floor — wearing stoles printed to resemble kaffiyehs, the black-and-white scarves that have become a symbol of Palestinians and bearing the words “Democrats for Palestinians” — many attendees have expressed strong support for a permanent cease-fire. But the official convention program has left her feeling “completely alienated,” she said, giving her “a feeling that the party wants to brush the issue of Gaza under the rug.”

Democratic leadership is out of touch with its base when it comes to the war in Gaza, Ms. DeReus argued. But she said she was hopeful that Ms. Harris would “say something that indicates a serious, tangible shift” from the Biden administration’s current policy of staunch support for Israel when she accepts the party’s nomination in a speech Thursday evening.

Rania Batrice, a Democratic strategist of Palestinian descent who was a deputy campaign manager for Senator Bernie Sanders of Vermont when he ran for president in 2016, said Ms. Harris had used “much more humanizing” language when speaking about the plight of the Palestinians, in what she described as a welcome change in rhetoric. The shift is meaningful to Ms. Batrice and her community, she said. “But that won’t stop the bombings in Gaza.”