The Houthi-run news agency Saba reported that the group, which said last month it would impose a naval blockade on Saudi Arabia in the Red Sea, attacked a Saudi ship carrying military equipment in the Bab el-Mandeb Strait. It did not name the ship, and there was no immediate Saudi response to the report.
The U.S. military, meanwhile, said a U.S. Navy MH-60 helicopter fired two Hellfire missiles to disable the steering gear of a Panama-flagged cargo ship.
The ship ignored repeated warnings to stop violating a naval blockade on Iranian ports, the U.S. Central Command said. Maritime sources told Reuters the ship was hit off Pakistan while sailing into the Gulf of Oman.
Both Iran and the United States have stepped up rhetoric in the past two days.
Iran's top security official, Mohsen Rezaei, said on Tuesday that the vital Strait of Hormuz shipping route will remain closed unless the U.S. accepts Iran's conditions to end the war – the release of Iran's frozen assets and an end to conflicts throughout the region, including in Lebanon and Gaza.
That followed a new demand from Trump on Monday that Iran should pay compensation for people killed in 50 years of wars, attacks and protests.
Throughout the conflict, Trump has alternated between threats of escalation and claims that a peace deal is imminent.
In an interview released late on Monday, he suggested the uncertainty could last for a while, saying he might just "bop along" and let Tehran fail economically, or hit them "really, really hard."
"I'm sort of negotiating," Trump told Real America's Voice. "They're very devious negotiators."
The comments from Rezaei, appointed on Sunday as second-in-command of the body that coordinates Iran's security and foreign policy, were the strongest indication the Strait of Hormuz would not reopen to shipping anytime soon. The waterway handled a fifth of global oil and liquefied natural gas flow before the war.
"As long as America does not change its behaviour and does not accept Iran's conditions, the Strait of Hormuz will not be opened," Rezaei said, according to the semi-official Tasnim news agency.
There was no immediate comment from Washington on Iran's latest comments.
Thousands of people have been killed in the conflict since the U.S. and Israel launched attacks on Iran on February 28.
Iran has struck U.S. assets and infrastructure in countries including Oman, Jordan, Kuwait, Israel, the United Arab Emirates and Saudi Arabia.
Raising the prospect of further escalation, Mohammad Reza Naqdi, an adviser to the commander of Iran's Revolutionary Guards, said on Iranian state TV on Tuesday that the corps was developing the ability to carry out operations "on enemy soil."
"We need to be able to move the operations to the enemy's soil, whenever this is needed and ordered," he said. "This is the characteristic of the offensive doctrine that has to be attained."
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