The BBC says it still has no independent verification of a claim by a Palestinian militant group that it has killed reporter Alan Johnston.
The corporation said it was "highly concerned" for Mr Johnston's safety and reiterated calls for his release.
On Sunday, the previously unheard of Tawhid and Jihad brigades claimed it had executed Mr Johnston.
BBC colleagues have rallied for Mr Johnston, 44, who was abducted at gunpoint in Gaza City on 12 March.
His parents, Graham and Margaret, say they are desperately worried and have appealed for anyone with information about their son to make contact.
Journalists have rallied in the Lebanese capital, Beirut, in support of Mr Johnston, as public campaigns to secure his release continue.
Born in Lindi, Tanzania, on 17 May 1962
BBC Kabul correspondent from 1997 to 1998
BBC Gaza correspondent from April 2004
Dozens of demonstrators held portraits of the journalist, who was seized at gunpoint in Gaza City on 12 March.
Others gathered outside the European Commission building in Brussels.
BBC colleagues held a vigil at London's Television Centre, in Birmingham and at Bush House, home of the World Service.
The corporation's Director General Mark Thompson said the BBC was still seeking clarification of the reporter's condition.
He praised Mr Johnston's family, calling the wait for news "agonising" for them.
Earlier, in a statement, the BBC said it could not confirm the claims by the Tawhid and Jihad (Holy War and Unity) brigades, describing the story as a rumour without verification.
"We continue to be highly concerned for [Alan Johnston's] safety and are demanding urgent clarification from the Palestinian and British authorities," it said.
There have been worldwide calls for Alan Johnston's release
The Tawhid and Jihad brigades made its claim in an email to media organisations, linking Mr Johnston's purported execution to the plight of Palestinian prisoners held in Israel.
A spokesman for UK Prime Minister Tony Blair said the government was investigating the reports.
"We are working closely with the Palestinian Authority and are urgently seeking information from them and other sources," he said.
In the latest of almost daily demonstrations in the wake of Mr Johnston's abduction, journalists gathered outside the Lebanese Press Syndicate building in Beirut to press for his release.
"We would like to appeal today to whoever is holding [Alan Johnston] to release him today unharmed," the BBC's Beirut correspondent, Kim Ghattas, told media.
Our correspondent said that before his abduction Mr Johnston had continued to do his job professionally, despite the risks.
But, more than a month since his abduction, there was concern about Mr Johnston's mental health and physical state, she said.
Another demonstrator, Lebanese TV reporter Diana Moukalled, said it was very important "to show our solidarity with any colleague that's harmed because of his job".
Earlier, Mr Johnston's parents Graham and Margaret Johnston issued a statement, saying it was a "desperately worrying time".
They added: "We make a heartfelt appeal to anyone who may have knowledge of Alan's situation and well-being to contact the authorities in Gaza."
Mr Johnston, from Scotland, joined the BBC World Service in 1991 and has spent eight of the last 16 years as a correspondent, including periods in Uzbekistan and Afghanistan.
He has lived and worked in Gaza for three years and was the only Western reporter permanently based in the often violent and lawless territory.
His posting in Gaza had been due to end in late March.