The Houthis have been targeting commercial vessels with drones and missiles in the Red Sea since mid-November, in what they describe as acts of solidarity with Palestinians against Israel in the Gaza war.
The group's military spokesman said it fired naval missiles at the Morning Tide and Star Nasia, identifying the Barbados- and Marshall Islands-flagged ships, respectively, as UK and US vessels.
The Greek-owned Star Nasia, managed by Star Bulk Carrier, was damaged by an explosion, a Greek shipping ministry official said, adding that its crew were not injured.
It is unclear whether the explosion was caused by a sea mine or a rocket, the official added.
The US military's Central Command said the Houthis fired three missiles at the Star Nasia, which reported minor damage but no injuries.
A US Navy ship operating near the Star Nasia shot down one of the missiles, Centcom said on X, formerly known as Twitter. It said the Star Nasia remained seaworthy and was continuing toward its destination.
Maritime security firm Ambrey said a Barbados-flagged, general cargo ship owned by a UK company suffered damage from an unmanned aerial vehicle while sailing south east through the Red Sea.
No injuries were reported.
The ship performed evasive manoeuvres and continued its journey, Ambrey said.
The owner of the Morning Tide, UK firm Furadino Shipping, told Reuters the ship was currently sailing without problems but gave no further information.
LSEG ship-tracking data showed the Morning Tide was sailing down through the Red Sea having come through the Suez Canal on Friday.
Its most recent signal shows it sailing out of the Red Sea through the Bab al-Mandab Strait.
The United Kingdom Maritime Trade Operations (UKMTO) agency had reported just after midnight GMT on Tuesday that a projectile had been fired at the port side of a ship located 57 nautical miles west of Hodeidah and that a small craft was seen nearby.
The projectile passed over the deck and caused slight damage to the bridge windows but the vessel and crew were safe and proceeded on the planned passage, UKMTO added.
The Red Sea attacks have disrupted global shipping and forced firms to re-route to longer and more expensive journeys around southern Africa and stoked fears that the Israel-Hamas war could spread to destabilise the wider Middle East.
The United States and United Kingdom began striking Houthi targets in Yemen last month in retaliation for the attacks on Red Sea shipping.
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