US President Donald Trump said on Tuesday that Iran had shot down a US Apache helicopter patrolling the Strait of Hormuz overnight and vowed to respond, deepening doubts about prospects for a truce announced in April in the Gulf conflict.

"I have just been informed by our Great Military that last night the Iranians shot down one of our highly sophisticated Apache Helicopters while patrolling over the Strait of Hormuz," Trump wrote in a social media post.

He said the two US pilots involved in the incident were safe and uninjured.

"Nevertheless, the United States must, of necessity, respond to this attack," he added.

The episode adds further strain to efforts to broker a peace deal to end the wider Middle East conflict and reopen Hormuz, a vital conduit for international trade in energy and other commodities.

Earlier on Tuesday, Trump said the two pilots were "fine" following their rescue, but made no comment about what caused the helicopter to go down.

A US Navy surface drone found and rescued the two crew members, the US military told Reuters. US Central Command said the AH-64 Apache went down at around 3 a.m. on Tuesday (2300 GMT on Monday).

Israel strikes Tyre, killing eight

On Monday, Israel and Iran said they would halt attacks on each other following an appeal by Trump to end their first direct exchanges of fire since April. However, Tehran warned it would resume hostilities if Israel continued to attack its ally Hezbollah in Lebanon.

In Tehran, two Iranian air defence personnel killed in Israeli strikes on Monday were due to be buried on Tuesday afternoon, Iran's military said. No deaths were reported in Israel following the Iranian strikes.

In a parallel conflict, Israel struck the historic port city of Tyre in southern Lebanon on Tuesday, killing at least eight people. It was the deadliest strike on the city since fighting erupted in Lebanon in early March, when Hezbollah launched rockets at Israel in support of Tehran after Israel and the United States began their war against Iran.

Israel had issued an evacuation order for the city earlier in the day. Residents fled and civil defence teams moved elderly residents to temporary shelters, state media reported.

The eight victims were killed in a single strike on the city's eastern edge, Lebanon's health ministry said.

A video verified by Reuters showed debris strewn across a road at the site of the attack.

Israel's refusal to end its campaign in Lebanon, as demanded by Iran, has complicated Trump's efforts to turn a fragile ceasefire in the wider US-Israeli conflict with Iran into a lasting settlement.

Trump told reporters earlier on Tuesday that he might have "an idea" for an Iran deal within a few days, without elaborating.

The Republican president, struggling with record-low approval ratings ahead of November's midterm elections, has repeatedly hinted that an agreement with Tehran was close, but no deal has yet materialised.

US and Israeli officials said Trump and Israeli Prime Minister Benjamin Netanyahu spoke on Monday.

In an interview with Axios, Trump said he had warned the Israeli leader against returning to war with Iran.

"I said, 'Bibi, you better be careful, or you will be on your own very soon,'" Trump said.

Israel warns of further strikes on Iran

However, Israeli military chief Eyal Zamir said on Tuesday that the attack Israel carried out against Iran the previous day was "in preparation for a much more significant and heavy blow".

"We are prepared to return and deliver another severe and deep strike against Iran," he said during a visit to training exercises in northern Israel.

Tehran has long maintained that any peace agreement with Washington depends in part on an end to fighting in Lebanon, which Israel invaded in March in pursuit of Hezbollah fighters who had fired across the border.

In northern Israel on Tuesday, Israeli troops operating in the Ramim Ridge area near the Lebanese border killed one person after returning fire during an incident, the military said.

Israel has never halted its Lebanon campaign, which has killed thousands of people, arguing that the conflict should be treated separately from any US-Iran ceasefire. Hezbollah has also continued its attacks.

At the same time, Tehran has continued to block most shipping through the Strait of Hormuz, which before the conflict carried around a fifth of the world's crude oil and liquefied natural gas. Washington has imposed its own blockade on Iranian ports.

US Energy Secretary Chris Wright said on Tuesday that ship traffic through Hormuz was rising "very meaningfully", but added that it would take many months for energy flows to return to normal after the conflict ends.

Trump has said any peace agreement must ensure that Iran cannot develop a nuclear weapon. Tehran's demands include the lifting of international sanctions, the release of billions of dollars in frozen assets and recognition of its control over the strait.