U.S. and Israeli interests in the ongoing Middle East conflict are diverging in both the short and long term, muddying the path to ending Israel’s war against militant group Hamas in the Gaza Strip.

Above all, Israel views Hamas as an existential threat and sees eradicating it as a crucial goal; anything short of that is a failure. The U.S. has committed to helping Israel defeat Hamas, but for President Biden, the threat goes beyond Hamas. His administration is trying to keep its allies united against Iran, Russia and China. Both countries want to avoid a larger regional war, but Israel is willing to take more risks in pursuit of defeating Hamas.

The ‘pause’ debate
In the immediate aftermath of the Oct. 7 attacks by Hamas, Biden made his staunch support for Israel clear, embracing Prime Minister Benjamin Netanyahu on a trip to Tel Aviv, a rare presidential visit to a war zone. But in the days since, Biden, under pressure from critics in his own party, has repeatedly stressed in phone calls with Netanyahu that Israel should run its military campaign in accordance with international humanitarian law. The U.S. is also increasingly calling for a pause in the fighting to get humanitarian aid into Gaza and hostages safely out, though resisting calls for a full cease-fire.

At a campaign fundraiser Wednesday, an attendee shouted, “As a rabbi I need you to call for a cease-fire right now.” Biden responded: “I think we need a pause.”

Secretary of State Antony Blinken said he discussed the issue and logistics of how a pause would work with Netanyahu and Israel’s war cabinet in a meeting in Tel Aviv on Friday.

In response to that pressure, Netanyahu said Friday he was opposed to a temporary cease-fire that didn’t include the release of Israeli hostages being held by Hamas. U.S. officials say Israel previously paused fighting when two American hostages were released. Israel agreed to pause airstrikes in the location where the hostages were being dropped off, according to a person familiar with the plan. Israel hasn’t acknowledged it agreed to the pause at the time.

The Biden administration is also pushing Israel to minimize civilian casualties and employ more surgical strikes targeting Hamas leaders. But even as the death toll climbs in Gaza and the humanitarian crisis worsens, Israel has continued large-scale strikes, including one at the Gaza Strip’s largest refugee camp that resulted in the deaths of women and children.

The White House has declined to weigh in on whether the strike was appropriate, but administration officials have grown frustrated with the wide-scale casualties.

“These are their operations and they—and only they—can speak to their targeting decisions and the way they’re conducting the operations,” White House National Security Council spokesman John Kirby said. “What we’re gonna do is make sure that they’ve got the tools and capabilities, including our perspectives, lessons that we learned in this kind of warfare as they venture into these operational decisions.”

Netanyahu said Friday he would continue his military campaign in Gaza with “all of its power.” He also said he wouldn’t allow fuel, which the U.S. and humanitarian groups say is needed to operate generators for hospitals and water facilities, to enter the Gaza Strip.

Gaza’s unclear future
In the long term, Biden is increasingly calling for a two-state solution, and Blinken has conferred with Israel’s government on what comes next for Israel and Gaza after its military campaign against Hamas.

“We are and will continue to have discussions with partners throughout the region and well beyond about what should follow once Hamas is defeated,” Blinken said Friday. “The best path, maybe even the only path, as I said, is through two states for two peoples.”

Israelis are unclear on what the end of the war would mean for Gaza’s future. Israeli Defense Minister Yoav Gallant has said the goal is to destroy Hamas’s military capabilities and its ability to govern, but added that his country has no interest in permanently reoccupying Gaza. And some on the Israeli far right have backed occupying Gaza in the longer term or pushing Gazans to the Sinai region of neighboring Egypt, which Egyptian leaders oppose.

Domestic pressures intervene
As the conflict continues with no clear end, Biden and Netanyahu both face growing political pressures at home.

Biden, who was initially praised for his support for Israel, is now seeing criticism from members of his own party—especially younger voters and Muslims and Arab-Americans—who are concerned about the growing death toll in Gaza and urging the administration to call for a cease-fire.

Even members of Congress who have been supporters of Israel and back military aid for the country said this week they wanted to see more restraint in its military campaign against Hamas.

“The current rate of civilian death inside Gaza is unacceptable and unsustainable,” Sen. Chris Murphy (D., Conn.), who sits on the Middle East subcommittee of the Senate Foreign Affairs panel, said Thursday. “I urge Israel to immediately reconsider its approach and shift to a more deliberate and proportionate counterterrorism campaign.”

A recent Gallup poll found that Biden’s job-approval rating among Democrats fell 11 percentage points in the past month to 75%—the lowest from his own party during his presidency.

Some Republicans, meanwhile, including 2024 GOP presidential contenders, are citing Biden’s call for a pause in Gaza as evidence that his support for Israel is softening.

And Netanyahu, who before the conflict was already unpopular with Israelis over his push for a controversial judicial overhaul, is now dealing with criticism over how the Oct. 7 attacks happened, his refusal to accept any responsibility for it and questions about whether he should resign.

The prime minister, who built a reputation as a security hawk, has said an investigation would have to wait until the conflict is over and the “only thing that I intend to have resign is Hamas.”

Write to Tarini Parti at tarini.parti@wsj.com