The cargo ship said it was hit close to Oman by a projectile, British navy agency UKMTO said, hours after Iran warned vessels against taking routes that it had not approved.
Two US officials told Reuters that Iran had fired on the ship while Iran's Persian Gulf Strait Authority, which it established to manage requests for ships to travel through the strait, said vessels outside routes it has set will not be guaranteed safe passage.
Four sources identified the ship as the Singapore-flagged Ever Lovely.
A security source said it was likely targeted by a drone.
There was no immediate comment from the US government.
US President Donald Trump warned earlier this month that if Iran did not honour an agreement aimed at ending the war and reopening the strait that the United States would probably go back to bombing the country again.
The IMO was helping to get hundreds of stranded ships and thousands of seafarers out of the strait where they had been stranded for months since the start of the war in late February.
It decided "to temporarily pause its implementation in order to reconfirm that the necessary safety guarantees continue to be in place for the ships on our evacuation list and all those in the region," IMO Secretary General Arsenio Dominguez said in a statement.
The IMO said the ship involved in the suspected attack was not part of its program.
The initiative, which was launched on Tuesday, was a voluntary option for ships and their crew to sail out of the Gulf using two routes – one via Iranian waters and the other via Omani waters, with US oversight, the IMO said this week.
Benchmark oil prices rose 1.9 per cent following the reported attack, which analysts said rekindled concerns about how long it could take for Gulf oil flows to resume normal levels.
The Oman incident is likely to refocus attention on the extent of Iran's future control over the Strait of Hormuz which, before the conflict, handled a fifth of the world's daily oil and liquefied natural gas supplies.
Before the incident, US Secretary of State Marco Rubio - wrapping up a tourof the Gulf to reassure states about the interim pact - told reporters that if Iran threatens or blocks ships in the strait "then we're going to have a problem".
Iran's Revolutionary Guards said on Thursday that safe passage through the strait would only be possible through routes designated by Iran, adding that it would take action against vessels that failed to comply.
The Revolutionary Guards also ordered two Panama-flagged ships to change course on Thursday, UK maritime security company Ambrey said.
Earlier, US Energy Secretary Chris Wright said shipments through the strait were approaching levels registered before the US and Israel launched strikes on Iran on February 28, with at least 20 million barrels of oil exiting the waterway in the previous 24 hours.
