Iran's semi-official Tasnim news agency posted a video of the ship anchored at sea, its name clearly visible.
The Stena Impero is British-flagged and the Mesdar is Liberian-flagged but British owned.
Iran has released video footage that it says disproves a claim made by the United States that a USA warship downed an Iranian drone near the Persian Gulf.
A statement from Stena Bulk, which owns the Stena Impero, said ship manager Northern Marine Management had lost contact with the crew of 23 after "unidentified small crafts and a helicopter" approached the vessel at about 4pm yesterday.
President Donald Trump insisted Friday that the American military had downed an Iranian drone that was threatening a USA naval vessel in the strategic Strait of Hormuz, which Tehran denied.
The action prompted a late night meeting of the COBRA emergency security committee chaired by Prime Minister Theresa May. "We are in close contact with both the United Kingdom and Swedish government authorities to resolve this situation and we are liaising closely with our seafarers' families", Hanell said. "It is essential that freedom of navigation is maintained and that all ships can move safely and freely in the region".
"We will respond in a way that is considered but robust and we are absolutely clear that if this situation is not resolved quickly there will be serious consequences", he said.
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Iran's top military spokesman also said there had been no encounter with any warship. "We are looking at a diplomatic way to resolve the situation".
Labour leader Jeremy Corbyn said: "The seizure of these vessels is unacceptable, and the tanker that remains under Iranian control must be released".
The United States will take advantage of the aggravation of the situation in the Strait of Hormuz to strengthen its military presence in the region, the chairman of the Federation Council's global affairs committee Konstantin Kosachev said.
The incident came as Iran and the United States emphatically disagreed on Friday over Washington's claim that a U.S. warship downed an Iranian drone near the Persian Gulf.
Britain's Royal Marines assisted in the seizure of an Iranian oil supertanker on July 4 by Gibraltar, a British overseas territory off the southern coast of Spain.
The tanker was taken to a coastal area and turned over to the authorities to take the necessary legal steps, state television said.
Iran seized at least one British oil tanker in a vital Persian Gulf waterway Friday, an escalation of tensions with the West that revived fears of a military clash, even as voices on both sides appeared to be seeking room for negotiations. Iran denied any attempted seizure.
The Trump administration has accused Iran of being behind a string of incidents, including attacks and harassment against commercial shipping, that have contributed to the rising tensions in the region as the United States sets about squeezing Iran with tighter sanctions.
Were the wider potential consequences adequately examined?
"But we are very clear that it must be resolved", he said, adding that he spoke with US Secretary of State Mike Pompeo about the incident and would soon speak with Iranian Foreign Minister Mohammed Javad Zarif.