Ten crew members have been rescued and at least three others killed after a cargo ship was attacked by Yemen's Houthis and sank in the Red Sea, a European naval mission says.

The Liberian-flagged, Greek-operated Eternity C was carrying 25 crew when it sustained significant damage and lost all propulsion after being hit by rocket-propelled grenades fired from small boats on Monday, according to the UK Maritime Trade Operations (UKMTO) agency.

The attack continued on Tuesday and search rescue operations commenced overnight.

The Iran-backed Houthis said they attacked the Eternity C because it was heading to Israel, and that they took an unspecified number of crew to a "safe location".

The US embassy in Yemen said the Houthis had "kidnapped many surviving crew members" and called for their immediate release.

Authorities in the Philippines said 21 of the crew were citizens. Another of them is a Russian national who was severely wounded in the attack and lost a leg.

The EU's naval mission in the Red Sea, Operation Aspides, said four more people were rescued on Wednesday night, included three Filipinos and one Greek citizen, bringing the total rescued to 10.

Greece-based maritime security firm Diaplous released a video on Wednesday that showed the rescue of at least five seafarers who it said had spent more then 24 hours in the water, according to Reuters news agency.

"We will continue to search for the remaining crew until the last light," Diaplous said.

Reuters also cited maritime security firms as saying that the death toll was four.

It is the second vessel the Houthis have sunk in a week, after the group on Sunday launched missiles and drones at another Liberian-flagged, Greek-operated cargo ship, Magic Seas, which they claimed "belong[ed] to a company that violated the entry ban to the ports of occupied Palestine".

Video footage released by the Houthis on Tuesday showed armed men boarding the vessel and setting off a series of explosions which caused it to sink.

All 22 crew of Magic Seas were safely rescued by a passing merchant vessel.

Since November 2023, the Houthis have targeted around 70 merchant vessels with missiles, drones and small boat attacks in the Red Sea and the Gulf of Aden.

They have now sunk four ships, seized a fifth, and killed at least seven crew members.

The group has said it is acting in support of the Palestinians in the war between Israel and Hamas in Gaza, and have claimed - often falsely - that they are targeting ships only linked to Israel, the US or the UK, which have carried out air strikes on Yemen in response.

The US state department condemned the attacks on the Magic Seas and Eternity C, which it said "demonstrate the ongoing threat that Iran-backed Houthi rebels pose to freedom of navigation and to regional economic and maritime security".

"The United States has been clear: we will continue to take necessary action to protect freedom of navigation and commercial shipping from Houthi terrorist attacks, which must be condemned by all members of the international community."

In a separate development on Thursday, Israel's military said its air force intercepted a missile launched from Yemen. It gave no further details.

In May, the Houthis agreed a ceasefire deal with the US following seven weeks of intensified US strikes on Yemen in response to the attacks on international shipping.

However, they said the agreement did not include an end to attacks on Israel, which has conducted multiple rounds of retaliatory strikes on Yemen.

The secretary-general of the International Maritime Organization (IMO) called for intensified diplomatic efforts following the new wave of attacks.

"After several months of calm, the resumption of deplorable attacks in the Red Sea constitutes a renewed violation of international law and freedom of navigation," Arsenio Dominguez said.

"Innocent seafarers and local populations are the main victims of these attacks and the pollution they cause," he warned.