The attacks are believed to be the latest in the Iranian-backed rebels' campaign that has disrupted the $US1 trillion ($A1.5 trillion) in goods that pass through the Red Sea each year over the Israel-Hamas war in the Gaza Strip as well as halted aid shipments to conflict-ravaged Sudan and Yemen.
Meanwhile, the efforts to salvage the still-burning Sounion seek to head off the potential ecological disaster posed by its cargo of one million barrels of crude oil.
In Monday's first assault, two projectiles hit the vessel, and a third explosion occurred near the ship, the British military's UK Maritime Trade Operations centre said.
"Damage control is under way," the UKMTO said.
"There are no casualties onboard and the vessel is proceeding to its next port of call."
The timing of the attack and co-ordinates offered by the UKMTO corresponded to the reported path of the Panama-flagged oil tanker Blue Lagoon I, now travelling south through the Red Sea to an unlisted destination.
The Blue Lagoon I was coming from Russia's port of Ust-Luga on the Baltic Sea and had broadcast it had Russian-origin cargo on board.
Later on Monday, the UKMTO reported a second attack off the Houthi-controlled port city of Hodeida.
The private security firm Ambrey said an aerial drone hit a merchant ship, though no damage or injuries were reported.
The attack happened only a few kilometres from where the Blue Lagoon I attack occurred, Ambrey said.
The Houthis did not immediately claim responsibility for the attacks.
However, it can take the rebels hours or even days to acknowledge their assaults.
The Houthis have targeted more than 80 vessels with missiles and drones since the war in Gaza started in October.
They seized one vessel and sank two in the campaign that also killed four sailors.
Other missiles and drones have either been intercepted by a United States-led coalition in the Red Sea or failed to reach their targets.
The rebels maintain that they target ships linked to Israel, the US or the UK to force an end to Israel's campaign against Hamas in Gaza.
However, many of the ships attacked have little or no connection to the conflict, including some bound for Iran.
Also on Monday, a salvage effort appeared to be under way to tow away the Sounion oil tanker, which was earlier hit by the Houthis and abandoned by its crew.
The Sounion was carrying some one million barrels of oil when the Houthis initially attacked it on August 21 with small arms fire, projectiles and a drone boat.
A French destroyer operating as part of the European Union's Operation Aspides rescued the Sounion's crew of 25 Filipinos and Russians, as well as four private security personnel, after they abandoned the vessel and took them to nearby Djibouti.
