Kay Coombes, the sister of Steve Warren, said that she and their mother, Margaret Warren, were convinced that he was still alive and the family was going through an “utter nightmare”.
The plea by the yachtsmen’s families to resume the search for the crew of the Cheeki Rafiki has been backed by an MP, prominent figures in the sailing world and an online petition supported by more than 20,000 people.
The 40ft yacht ran into difficulties around 620 miles east of Cape Cod in Massachusetts on Thursday while returning to the UK from a regatta in Antigua.
Contact with the ship’s experienced captain Andrew Bridge, 22, and crew members James Male, 23, Steve Warren, 52, and Paul Goslin, 56, was lost in the early hours of Friday while they diverted to the Azores.
US and Canadian aircraft assisted by three merchant vessels looked for them throughout Friday and Saturday but called off the search on Sunday at 5am local time amid treacherous weather.
Some 4,000 square miles were scanned for the “very well-equipped” vessel’s two personal location GPS beacons until no more transmissions were received from the devices, which have a short battery life.
On Saturday, a cargo vessel helping with the search spotted and photographed an overturned hull which matched the description of the Cheeki Rafiki but reported no signs of people on board or a life raft.
Kay Coombes said: “It’s an utter nightmare, we are grateful for the US and Canadian coast guards for what they have done so far but it’s stopped too soon after two days, it’s not long enough, we believe they are still alive.
“They are four strong-minded, physically strong sailors, they knew they were in difficulties and had every opportunity to get into the life raft which would have had provisions for several days. But if no one is looking for them, they won’t be found.”
Mr Bridge’s aunt, Georgina Bridge, said: “Obviously we are all devastated by what is happening. We just want the search to resume.
“We have great appreciation for the efforts the US Coastguard and our Foreign Office have made but obviously we believe there is hope and there is a possibility of them being found alive.
“The hull has not been examined so that is a possibility, there is a possibility they may have been able to launch a life raft, we just do not know.
“Obviously the conditions are difficult; we appreciate every effort is being made but we were surprised the search was called off so early. We are in contact with the Foreign Office and they have been helpful.”
Caroline Nokes, the Conservative MP for Romsey and Southampton North, said that she had been told by the Foreign Office that it was “investigating every avenue it can do” to try to encourage the US Coastguard to act for longer.