It issued a statement after an anti-piracy maritime group insisted it should be allowed to negotiate a payment for Paul and Rachel Chandler.
The foreign secretary said political and diplomatic efforts were ongoing.
The Chandlers, of Tunbridge Wells, Kent, were captured while sailing towards Tanzania on 23 October.
The latest news of the couple emerged on Sunday. They spoke of their ordeal through a news agency which had been allowed to send a photographer when they were examined by a doctor last week.
The Chandlers said they were being badly treated and were in urgent need of help.
The pirates have threatened to kill the couple if their demands for $7m (Ã‚Â£4.4m) are not met.
Nick Davis, chairman of the Merchant Maritime Warfare Centre, which provides anti-piracy advice and training, said the Chandlers’ captors were running out of patience.
“The door is open for them to be released. Somebody needs to pick up the gauntlet and run with it,” he said.
“For the amounts involved, I don’t think it’s worth trying to bring anyone to justice. We just need to get Paul and Rachel home.
“We are the people who know what needs to be done, we can do it, we just need to be allowed to do it.”
Mr Davis said he could not do anything without the “green light” from the Foreign Office and the Chandlers’ family.
He said the pirates would need to feel they were talking to somebody with the authorisation and means to secure the release.
A Foreign Office spokesperson said it had “never attempted to block any activity by Mr Davis”.
“Although there is no UK law against third parties paying ransoms, we counsel against them doing so because we believe that making concessions only encourages future kidnaps.
“This is why the government does not make or facilitate substantive concessions to hostage takers.”
Speaking in the Commons, Foreign Secretary David Miliband said the latest pictures of the couple had been “heart-wrenching”.
“The political and diplomatic effort continues in close liaison with the family,” he said
“Everybody’s hearts goes out to the Chandlers and we will continue to make every effort to help resolve this very distressing case”.
The couple were examined by a surgeon on Thursday and Mrs Chandler was reportedly in a poor state of mental and physical health.
Mrs Chandler, who was being held in a separate location from her husband, told the news agency AFP: “I’m old, I’m 56 and my husband is 60 years old. We need to be together because we have not much time left.”
Mr Chandler appeared in better health than his wife but was also under extreme stress, AFP said.
“I just want to say please, to my government, get me and my wife out of here,” he said.
“We have no money and we can’t pay a ransom. We just need the government to help, anyone who can help us out of here.”
A Downing Street spokesman said the government was in regular contact with the Chandler family.
“The prime minister’s view is that hostage taking is never justified and we would like to repeat the government’s call that Paul and Rachel Chandler should be released immediately and unconditionally,” he said.
“We are doing everything we can to secure their release.”