The parents of BBC Gaza reporter Alan Johnston have said they are desperately worried after Palestinian militants claimed to have killed their son.
In a statement, the BBC said it could not confirm the claims by the Tawhid and Jihad brigades, describing the story as a rumour without verification.
In a separate statement, Graham and Margaret Johnston appealed for anyone with information to make contact.
Journalists rallied in Beirut, Brussels and at BBC premises to show support.
In Beirut 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.
Born in Lindi, Tanzania, on 17 May 1962
BBC Kabul correspondent from 1997 to 1998
BBC Gaza correspondent from April 2004
He praised Mr Johnston's family, calling the wait for news "agonising" for them.
On Sunday the Tawhid and Jihad brigades (The Brigades of Holy War and Unity), a previously unheard of group, said it had killed Mr Johnston, linking his purported killing 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 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.
"Because we, ourselves, have suffered a lot in Lebanon, we have a long history of targeting journalists, some of them were killed, others kidnapped, some of them were tortured.
"It's unfortunate for journalists to be targeted just because they are doing their 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.
"Our son has lived and worked among the people of Gaza for the last three years to bring their story to the outside world and we ask every one of them to help end this ordeal."
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.