"We both share a desire to see them returned safely to their families," Brown said in a statement.
"There are many people working behind the scenes trying to find a solution."
A foreign office spokewoman said the ministry had no "independent verification of the claims in this video" and that she could not comment on its accuracy.
David Miliband, the British foreign secretary, said in a statement that the latest communication from the hostage-takers "will cause deep distress and concern to the families of the five men".
He said the UK officials in Iraq "continue to be ready to work with anyone prepared to help in this case".
Mowaffak al-Rubaie, Iraq's national security adviser, had said in June that the five Britons were still alive.
The video from a group calling themselves the Shia Islamic Resistance in Iraq, posted on the newspaper's website, begins with the photo of a man appearing alongside the Arabic text of the statement.
According to the English translation provided by The Sunday Times, it claims that the hostages had slipped into depression and made "more than one attempt at suicide".
|The captors had originally demanded that the UK pull its troops out [AFP]
It goes on to say that Jason blamed "procrastination, and foot-dragging, and lack of seriousness on the part of the British government".
The Sunday Times said the video was handed to one of the newspaper's representatives by an intermediary, who said that proof of the victim's death would only be provided if the British government agreed to negotiate.
The statement called for the release of nine prisoners being held in Basra in return for the freedom of the hostages.
It then shows video footage of a second man, named by The Sunday Times as Alan, who said on camera that while he had been treated "very good, to say the least", he was "not doing well" physically. He said: "Psychologically, I'm doing a lot worse."
Alan called on the British government to "please hurry" and "get this resolved as soon as possible".
The men - an IT consultant and his four bodyguards - were seized at the Iraqi finance ministry in May 2007.
The computer expert has been named as Peter Moore, from the eastern English city of Lincoln, who was working for US management consultancy BearingPoint.
The identity of the other four men has not been revealed, although it is known they were employed by Canadian security firm GardaWorld to guard Moore.
The Shia Islamic Resistance in Iraq has previously released two videos of their captives.
The first video featuring one of the captors was released in December.
It included a statement from the captors threatening to kill him unless the UK pulled its troops out of Iraq.
A second video was released in February, this time featuring a second hostage who appealed for the release of nine Iraqis in return for the Britons' freedom.