In a statement, the coastguard said the ship participated in the search before docking in Santa Cruz de Tenerife in the early hours of Friday.
Rescue boats, two helicopters and a specialist search and rescue aircraft have been deployed to the area.
Lesley-Anne Kelly, a passenger on the ship, recalled the moment an alarm sounded while she was having breakfast with her mother on Thursday morning.
She said the alarm was followed by an announcement of “man overboard”.
Mrs Kelly said the captain told passengers the ship had to stay in place until the coastguard allowed it to leave, and later announced that the search had been stood down due to darkness.
“It was pretty sombre last night, especially after the announcement that they were standing down the search,” she said.
Passengers were told that witnesses were being interviewed, she added.
Mrs Kelly said she believed people “had clearly seen it happen”, adding: “It was breakfast time. Everybody was up and about. It was pretty bright, so yeah I can imagine if someone had gone in at that time of the day they would have been seen by multiple people.”
