Iran and Oman have been in on-and-off talks for weeks about controlling traffic through the strategic waterway, which handled one-fifth of global oil and liquefied natural gas shipments before the war began in February.
Most shipping has been shut down since then.
The two countries said on Tuesday they discussed "a joint temporary navigational corridor" through the strait and agreed to clear it of mines.
Trump said all mines in the strait had been cleared, repeating comments that he has previously made.
Though active hostilities between the US and Iran have largely subsided, diplomatic efforts to reach a peace deal have stalled and passage through the waterway remains perilous.
An oil tanker was struck on Tuesday by an unidentified projectile and disabled about nine nautical miles northeast of Oman's Ash Shishah, which lies at the entrance to the strait, the United Kingdom Maritime Trade Operations said.
Separately, the US Secret Service said on Tuesday that it was aware of a video broadcast by Iranian state television discussing a potential plot to assassinate Trump's youngest son, Barron.
The three-minute video discussed a potential plot to assassinate Barron Trump, 20, claiming he was being monitored and alleging a $US10 million ($A14 million) bounty had been offered for his killing.
"The US Secret Service is aware of the video and investigates anything that can be perceived as a threat toward our protectees," Secret Service spokesman Nate Herring said in a statement.
"Out of concern for operational security, we do not discuss matters of protective intelligence."
Since the US assassination of Iran's Ayatollah Ali Khamenei, Iranian media have on multiple occasions circulated content threatening the president and family members.
Despite the ongoing tensions, the US is beginning to send personnel back to some diplomatic missions in the Middle East that were evacuated or downsized amid tensions with Iran, two people familiar with the matter told Reuters.
The move suggests Washington sees a lower risk of the conflict with Iran escalating in the near term, though some embassies will initially operate below full capacity.
In an effort to hobble Iran's economy, the United States earlier this week threatened to punish countries that continue to do business with Iran, but said it would not impose penalties immediately.
Oil prices fell for a second day as traders brushed off the impact of the US sanctions, while Iran denounced the US effort to isolate its economy as an act of "gross lawlessness", expressing confidence that many countries would not join the pressure campaign.
US public approval of the war has fallen to its lowest level since the conflict's early days, and Trump's popularity is likewise at a record low ahead of congressional elections in November, according to Reuters/Ipsos polling.
Thousands of people have died in the conflict, most of them in Iran and Lebanon, while the US and Israel have degraded much of Iran's conventional military capacity.
The exact state of its nuclear program, which the US and Israel aim to dismantle, remains unknown.
