TEL AVIV—President Trump’s deal to wind down the war with Iran set off alarm bells in Israel, where top officials are wrestling with the consequences of easing the pressure on Tehran and the risks of opening a rift with the U.S. over the war with Hezbollah in Lebanon.

The tension has been heightened by the lack of certainty about what exactly Trump has agreed to in the deal, which is expected to be signed later this week. Israeli Prime Minister Benjamin Netanyahu was urgently trying to set up a meeting with the president to sort out the competing issues, a person familiar with the matter said.

Trump criticized the strike in an interview with The Wall Street Journal and said on social media that Israel had to stop its attacks across Lebanon. That was at odds with a U.S.-brokered ceasefire agreement earlier in June that only required Israel to end the fighting if Hezbollah also stopped. Iranian Foreign Minister Abbas Araghchi said the U.S. was on the hook to end Israel’s attacks and aggression in Lebanon, state media reported.

Defying those claims, Israeli Defense Minister Israel Katz said the military would hold its so-called security zone in Lebanon indefinitely, saying it was needed to protect communities in northern Israel. He also said Israel would act independently to stop Iran from getting nuclear weapons if necessary.

At a news conference Monday, Netanyahu declined to criticize the emerging U.S.-Iran deal and said his country’s war aims in Iran had largely been achieved. He said Iran’s war-production capacity had been damaged, its economy was in tatters, and its nuclear program set back.

“People ask what we have achieved, and the answer is: we have pushed away the immediate threat of annihilation,” Netanyahu said. “The struggle is not over and done. We will need to continue to stand guard and defend ourselves as necessary.”

Privately, Israeli officials are concerned that Trump has agreed to a deal that could provide Tehran with the financial relief it needs to rebuild its economy but doesn’t include a commitment to turn over its enriched uranium.

It is a harsh comedown from Israeli hopes that the war would bring fundamental change to the region by toppling or crippling the Iranian regime and paving the way to diplomatic relations with more of Israel’s Arab regional counterparts under an American security umbrella, said Michael Oren, a former Israeli ambassador to Washington.

“If Iran receives sanctions relief of billions of dollars in assets, it will rebuild its military capabilities and proxies, and the blow to U.S. prestige in the region will be immense if not irreversible,” Oren said.

Many analysts saw those hopes as unrealistic from the start. Netanyahu is now coming under attack from across the political spectrum from those who say he led the country into a misguided war and mishandled the relationship with the U.S.

“Israelis are deeply disappointed in this outcome, but they should not be surprised,” Daniel Shapiro, a former U.S. deputy assistant secretary of defense for the Middle East, said on social media.

The U.S. and Israel started the war with unusually tight alignment. American and Israeli war planners still sit side by side and dozens of American tankers are parked at airports ready to refuel Israeli jet fighters. Trump and Netanyahu both said the war would give Iranians the opportunity to rise up and take overthrow the regime.

But their interests quickly diverged after Iran shut the Strait of Hormuz and choked off oil and gas supplies. Trump has moved to end a conflict that wasn’t popular with American consumers at the same time as Netanyahu was trying to raise the pressure on the Iranian regime. Both leaders face elections in the fall, compounding the tensions.

Oren said Israel now has its back against the wall and can’t make concessions in the fight against Hezbollah even if it risks its relationship with Trump.

“It’s not a question of what Israel can do, but what it must do,” he said. “There’s almost no wiggle room there.”

While Israel would have lobbied Congress in the past to influence the White House, Israel no longer holds that kind of sway among lawmakers or the American public, said Oren.

Some former Israeli security officials said Netanyahu was now clashing with Trump due to domestic pressures from his political base just a few months before an Israeli election. They also said Israel should have refrained from striking Beirut while it was so politically sensitive. Netanyahu’s right wing base had strongly advocated for striking Beirut in response to any attacks on northern Israel by Hezbollah.

“He is playing pure politics. Bibi in the end is all about Bibi. He’s desperate that he’ll lose the elections,” said Chuck Freilich, a former deputy national security adviser, referring to Netanyahu by his popular nickname. “Clashing with Trump is the last thing Israel should do.”

While the U.S.-Iran agreement isn’t good for Israel, the damage done to Iran by the U.S. and Israel in two wars over the past year has severely weakened Iran both economically and militarily, and left Israel in a much better position than if the wars hadn’t been waged at all, said Erez Winner, head of the Israel Defense and Security Forum.

“I don’t think anyone is regretting the wars, but maybe we’ll be sorry we didn’t do more,” Winner said.

Before the war’s outset, Israel and the U.S. had demanded the dismantling of Iran’s nuclear program, caps on its ballistic missile program and the end of its support for allied militant groups around the region, including Hezbollah in Lebanon, and Hamas and Islamic Jihad in Gaza.

The deal Trump and Iran have agreed to is a narrow memorandum of understanding aimed at opening the strait to maritime traffic, putting in place an extended halt to the fighting and setting up later talks on the difficult issues of Iran’s nuclear program and sanctions relief.

U.S. officials say Iran will only receive financial relief in return for concrete progress on American demands. Iranian state media has said the country will get access to some of its cash currently frozen overseas in the near term.

Avner Golov, a former Israeli national security official, said Trump’s deal risks legitimizing the Iranian regime and surrendering the threat of American force, while allowing Iran to threaten the strait. “It’s not a good starting point for negotiations,” he said.

Israeli officials are also worried that Iran will be able to either draw out the nuclear negotiations indefinitely or get U.S. agreement on a deal that will leave it with the components required to rehabilitate its nuclear program in the future.

Trump told the Journal on Sunday that Israel shouldn’t be concerned.

“Bibi is OK with it,” he said. “Why is it good for Bibi? Iran cannot have a nuclear weapon under any circumstance.”

Israeli analysts point out that the Obama administration took years to hammer out the intricate details of the nuclear accord that Trump withdrew from in his first term.

Iran still has tons of enriched material, including enough near-weapons-grade material to fuel almost 11 nuclear weapons. It also has an unknown number of centrifuges used for enriching uranium.

Ariel Kallner, a lawmaker from Netanyahu’s Likud party, said in a social-media post Sunday that the deal shows Israel can’t rely on the U.S. for its security.

“The problem is that parts of it are still seeking easy and quick victories, or agreements with nice promises—momentary quiet instead of striving for victory, even if there are costs in the present,” he said.

Israel, however, has relied heavily on American munitions, equipment and missile defenses over the past three years of war.

While the war launched in February piled new damage on Iran’s economy and killed a number of top leaders, it hadn’t prevented the regime from responding quickly to provocations with military force or disrupted its iron grip on the population. Many analysts say further attacks were unlikely to change that equation.

Israeli security officials felt the U.S. blockade of Iranian ports had succeeded in squeezing the regime and wanted to see it continue. Some American hawks shared that view, but Trump was worried about the squeeze on American consumers.

Trump’s decision to do the deal instead showed the limits of Israel’s influence over American policy decisions it doesn’t like.

“Trump will do what he thinks is good,” Michael Herzog, a former Israeli military officer and ex-ambassador to the U.S., said Sunday on Army Radio, “whether it’s good or not good for Israel.”

Write to Dov Lieber at dov.lieber@wsj.com