President Donald Trump said the U.S. and Israel are making significant progress in the war on Iran and could end the conflict “very soon,” cooling a surge in oil prices.

Trump said the operation that began Feb. 28 is ahead of schedule, while cautioning that fighting would still not be over this week. The two allies continued to bomb Iran overnight and into Tuesday morning, with the Islamic Republic firing drones and missiles at targets across the Middle East in return.

Asked whether he would be willing to talk to the Iranians, Trump said it would be possible, Fox News reported on Tuesday. Iranian officials have pushed back against that idea, with Parliament Speaker Mohammad Bagher Qalibaf saying on X that the country is “absolutely” not seeking a ceasefire.

Foreign Minister Abbas Araghchi told PBS’s News Hour on Monday that talks with the U.S. are not “on our agenda.”

Addressing concerns about soaring energy prices, Trump said Monday that the U.S. Navy will escort tankers to maintain a steady oil supply through the Strait of Hormuz. The effective closure of the waterway — vital to the world’s flow of petroleum — has created bottlenecks and caused regional energy giants to slash production.

Brent crude, having climbed to almost $120 a barrel early Monday, eased to about $91 in Tuesday trade. That’s still more than 50% higher than at the start of the year.

Saudi Arabia, Iraq, the United Arab Emirates and Kuwait have lowered their collective output by as much as 6.7 million barrels a day, people with knowledge of the matter said. That amounts to about 6% of global supply. There’s little sign Hormuz can be opened quickly without at least a pause in hostilities.

Saudi Aramco Chief Executive Officer Amin Nasser called the disruption “the biggest crisis the region’s oil and gas industry has faced.”

Trump told Republican lawmakers that the U.S. and Israel are “crushing the enemy in an overwhelming display of technical skill and military force.” He later said the U.S. had hit 5,000 targets in Iran, reducing the country’s missile capability to a 10th of what it was. The military objectives of the war could be described as “pretty well complete,” he said.

Even so, Iran pressed ahead with attacks on Israel and Arab states on Tuesday. The United Arab Emirates said it was responding to a missile threat from the Islamic Republic, hours after it reported a drone attack on its consulate in Iraq’s Kurdistan region. One person was killed in an attack on the Bahraini capital of Manama, while Kuwait intercepted six drones.

There are early signs the attacks on some countries, including the UAE, are waning. Countries from Australia to South Korea and the U.K. have either helped to boost defenses of Gulf allies or are assessing requests to do so.

Iranian President Masoud Pezeshkian said Tehran is prepared to scale back the conflict “on the condition that the airspace, territory, and waters” of neighboring countries aren’t used to launch attacks on the Islamic Republic, according to semi-official Mehr news agency.

Pezeshkian made those remarks during a phone call with Turkey’s President Recep Tayyip Erdogan, following the interception of an Iranian missile in Turkish airspace by NATO’s air defenses on Monday. He’s made similar comments in the past week, yet Iran continues to attack the likes of the UAE and Saudi Arabia, which say they aren’t allowing Israel or the U.S. to use their airspace for strikes.

NATO is increasing air defenses in southeast Turkey, where the U.S. operates a key radar supporting the alliance’s ballistic missile shield.

More than 1,300 Iranians have been killed in the war so far, according to an official toll that number hasn’t been updated for several days. The U.S. reported its seventh casualty on Sunday. Two Israeli soldiers and about a dozen civilians also have died, while several deaths have been reported in Gulf countries.

Israeli forces maintained attacks on southern Lebanon on Tuesday, aiming to degrade Iran-aligned Hezbollah. Some 486 people have been killed in Lebanon, according to the nation’s health ministry.

Trump pledged during his election campaign not to allow the U.S. to become involved in protracted foreign wars, and there’s a risk that more American casualties and sustained high gasoline prices will weigh on the Republican Party’s chances in mid-term elections in November.

Polls show an overwhelming majority of Americans oppose the war, and it’s been condemned by several GOP politicians.

Trump said he could waive “certain oil-related sanctions to reduce prices” but didn’t offer specifics beyond acknowledging he had discussed the topic with Russian President Vladimir Putin. Moscow has faced a range of restrictions on its vast oil industry, including a price cap on its crude and U.S. sanctions on its two largest producers — a bid to deprive the country of revenue over its war in Ukraine.

“We’re looking to keep the oil prices down,” the U.S. president said. “They went artificially up because of this excursion.”

On Sunday, Iran chose Mojtaba Khamenei as the country’s new supreme leader. His father Ali Khamenei ruled the country for almost 37 years and was killed when U.S.-Israeli strikes began on Feb. 28.

The 56-year-old has deep ties to the Islamic Revolutionary Guard Corps, about the most powerful military and economic organization in Iran. The group pledged full obedience to the new leader. He’s yet to speak publicly since being elected.

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—With assistance from Omar Tamo, Galit Altstein, Dan Williams and Paul Wallace.

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