Security firms involved in the operation said four crew members and one armed guard from the vessel had been pulled from the water, where they had been for more than 24 hours.
They said they had not yet made visual contact with the rest of the 22-member crew nor two more guards who were aboard.
"We will continue to search for the remaining crew until the last light," said an official at Greece-based maritime risk management firm Diaplous on Wednesday.
"We aim at a peaceful operation," the official said earlier.
Eternity C was first attacked on Monday afternoon with sea drones and rocket-propelled grenades fired from speed boats by Houthi militants, maritime security sources said.
Two security sources told Reuters on Wednesday that the vessel was attacked again on Tuesday night, forcing the crew to jump into the water.
The Iran-aligned group has not claimed responsibility for the attack.
One source said that there were fears some of the crew might have been kidnapped by the Houthis.
The crew is made up of 21 Filipinos and one Russian.
The four armed guards include one Greek national.
The Liberia-flagged, Greek-operated vessel's operator, Cosmoship Management, has not responded to requests for confirmation of casualties or injuries.
Greek government officials have been in diplomatic talks with Saudi Arabia, a key player in the region, over the incident, according to sources.
Eternity C is the second vessel attacked in the Red Sea in the past week following an assault that sank the cargo ship MV Magic Seas, after months of calm in the region.
The deaths are the first fatalities involving attacks on shipping in the Red Sea since June 2024.
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