Teams searching for the men earlier said they had found a body.
The missing men are 49-year-old skipper Martin Gardner, Edward Gardner, 50, and Ian Donald, 55, all from Fife, and Sidney Low, 52, from Aberdeen.
The alarm was raised when RAF Kinloss picked up a distress call from the Kirkcaldy-registered Meridian on Thursday night.
Norwegian rescue services said they had recovered one body near to where the boat disappeared.
Pieces of debris such as buckets, fishing equipment and boots were earlier discovered by a search team in the area.
The vessel was involved in guard duty around North Sea pipelines - making sure other vessels do not stray too close.
The Meridian, contracted by Talisman Energy (UK) Limited, had two life rafts on board and the crew were kitted out in survival gear.
It had been out since 11 October and had been due to complete its duties in four days' time.
An Aberdeen Coastguard spokesman said: "A body has been found with no ID and with no survival suit.
"That's all the details we have at the moment."
Rescuers said an empty life raft and debris had been found in the area, about 160 miles east of Aberdeen. The trawler's second life raft is still missing.
Anstruther resident
The vessel's owner Martin Gardner, from Anstruther, is married with children.
Edward Gardner, who is not related to the skipper, lives in Cellardkye but is a lifelong Anstruther resident.
Married with three children, he has been a fisherman for 35 years.
Mr Donald is another lifelong Anstruther resident who has also been a fisherman for 35 years.
The grandfather lives with his long-term partner in Pittenweem.
His sister, Ruth Guthrie, 60, told the Daily Record: "I have given up all hope."
She added: "The only comfort I have is the knowledge that they are all together."
Supt Bill Lawrie, of Fife Constabulary, said the force's officers and local councillors were providing support to the families and community.
"Anstruther and the surrounding villages are close-knit communities with a long seafaring tradition," he said.
"We are confident that these strong community ties will help its people through the days ahead."
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