The Qatari liquefied natural gas tanker Al Rekayyat reported that it had been struck overnight and its engine room set on fire.
Maritime security sources said on Tuesday that a Saudi crude oil tanker had also been damaged.
"Mayday mayday mayday. This is vessel Al Rekayyat, LNG vessel Al Rekayyat. We are being hit by drone on port side, top of engine room," the Rekayyat's captain said in a recorded radio call reviewed by Reuters.
He said the crew were safe but the ship had been disabled without engines or steering, and called for help from any vessels in the area.
There was no claim of responsibility for the attacks.
News website Axios reported that Iran had fired on two ships.
The incidents, the first reported attacks in the strait since mourning for Iran's supreme leader began on July 3, were a reminder that Gulf shipping remains unresolved more than four months after the US and Israel launched a war they said would stop Iran from being able to threaten its neighbours.
Iran's clerical rulers have exerted new-found control over the world's most important energy shipping route, where they aim to install a permanent system to collect fees in what would amount to a huge shift of the balance of power in a region where Washington has acted as guarantor of security for generations.
Iran's leadership has demonstrated its firm grip on the country during a week of mourning for Khamenei, who was killed along with his daughter, granddaughter, son-in-law and daughter-in-law on the first day of the war.
The caskets of the slain supreme leader and four members of his family were driven through the streets of the seminary city of Qom on Tuesday, where many hundreds of thousands of people carried flags and banners, and in chants vowed to avenge Khamenei.
Some bore signs reading "KILL TRUMP".
A similar huge funeral procession was held in the streets of Tehran on Monday, following more solemn prayer events that began on Friday, attracting top figures in Iran's leadership and dignitaries from abroad.
Authorities say the leader's body will be taken to Shi'ite holy cities in neighbouring Iraq, then brought back to Iran and laid to rest in a medieval shrine.
The war has been paused under an interim peace deal reached in June, intended to provide a 60-day period for negotiations on a permanent deal.
A round of indirect talks in Qatar concluded on July 2 with no sign of headway towards a lasting peace four months after the United State and Israel began the conflict.
Trump has repeatedly threatened to resume bombing, most recently on Monday when he told reporters in the Oval Office: "We're either going to make a deal or we're going to finish the job. OK. And it won't be tough to finish the job. I'd rather make a deal, because I don't want to affect 91 million people."
Iranian Foreign Minister Abbas Araqchi said such threats violated the terms of the memorandum of understanding reached to pause the war.
"Negotiations on final deal will not commence if threats continue," he wrote.
"Honour your signature."
Despite five days of mourning, there has still been no sign in public of his son and successor, Ayatollah Mojtaba Khamenei, believed to have been disfigured by wounds in the same attack and yet to be shown in any image since the war began.
Three other sons of the slain leader prayed at the casket on Sunday.
