A rescue mission was under way to locate Paul and Rachel Chandler, from Tunbridge Wells, Kent, after they went missing while sailing from the Seychelles towards Tanzania.
A news agency said it had been contacted today by a pirate to say the couple were "in our hands now".
The man, who gave his name as Hassan, said the captives were in good health and ransom demands would follow.
The Foreign Office said it was "urgently investigating" the claims.
"We are urgently investigating reports that Paul and Rachel's yacht has been seized and they are now being held captive," said a spokeswoman.
Leah Mickleborough, the couple's niece, said the family were alerted to an alarm signal made from the couple's yacht - the Lynn Rival - on Friday but initially thought the situation was not serious.
"We were just waiting for them to come into docks because it tended to be the case that, out in the middle of the Indian Ocean, it was quite difficult to get hold of them and it was expected that they would dock at a little island," she told Russell Fuller on BBC Radio 5 Live.
She added: "When you hear of things like this you possibly expect the worst might have happened but you always hope that it hasn't ... Things are turning out that potentially this might be happening.
"You never believe it's going to be one of those things that happens to your family."
Ms Mickleborough, who last saw the couple five weeks ago when they attended her wedding, said sailing was their passion.
"This is their life, really. They do sailing, they live for this," she said.
"They are not naive. They are very experienced in these things. They are not the sort of people who would put themselves deliberately in danger."
She added: "All of us as a family are extremely upset by what has happened. We are extremely distressed and it's such an emotional thing and such a horrible thing to be experiencing.
"I just hope they're ok. We all hope they are and that it can be resolved easily."
The route would have taken Mr and Mrs Chandler, aged 58 and 55, near Somali waters notorious for pirate attacks on ships and smaller boats.
The last message on their extensive travel blog was posted on Friday morning and reads simply: "Please ring Sarah."
It is thought the message was to Mrs Chandler's sister, believed to live in the London area.
Britain's Maritime and Coastguard Agency said an Emergency Position Indicating Radio Beacon (EPIRB) was activated at 11pm on Friday.
A spokesman for the coastguard in the Seychelles said: "They had been headed for Tanzania but we haven't heard from them. We don't have any information regarding them.
"There have been reports that they were hijacked by pirates but no one can prove that. We don't know what has happened and cannot speculate."
On their blog, the couple wrote on Wednesday that they were saying farewell to the Seychelles in the Indian Ocean - and were bound for Tanzania more than 500 miles away.
"We'll be at sea for 8 to 12 days, maybe 14 as we are now getting into the period of transition between the south monsoon and north monsoon, so the trade winds will be less reliable and we may get more light winds," they said.
"We probably won't have satellite phone coverage until we're fairly close to the African coast, so we may be out of touch for some time."
Elsewhere on the blog, the couple - who have been sailing around the world for several years - wrote of "the Somali pirate problem" that delayed other voyages to Tanzania.
Enthusiasts on a yachting forum questioned the wisdom of sailing in that area.
One post said: "I hope the people in question are well. However, I do wonder why anyone would sail through these areas. The dangers are very well known and very well reported."
International naval search teams are involved in the search.
A spokesman for the European Union Naval Force Somalia said: "Our ships will be scouring the seas on their normal passage and monitoring the airwaves."
He said the yacht had gone missing in "fairly dangerous waters" but there had been no confirmation it had been hijacked by pirates.
A Royal Navy spokesman said HMS Cumberland was one of the ships involved in the piracy crackdown.
The Foreign Office said relatives of Mr and Mrs Chandler were being kept informed.
Charlotte Dolwin, Mr and Mrs Chandler's cousin, told Channel 4 News: "They know the sea well. They wouldn't do anything they thought would be risky. They're not on a mission to do anything - they're just travelling around. I'm very surprised and very worried for them. I just hope that they return safely."
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