A Palestinian group calling itself the Tawhid and Jihad brigades has issued a claim that it has killed BBC Gaza correspondent Alan Johnston.
Alan Johnston: Praised for his calmness and courage
The BBC says it is aware of the reports and is deeply concerned, but stresses there is no independent verification.
In a statement faxed to news agencies, the group contrasts the attention given to Alan Johnston's captivity with that given to Palestinians held in prison.
Mr Johnston was abducted as he returned home from his Gaza office on 12 March.
The BBC has issued a statement saying it is deeply concerned about what it is hearing.
"But we stress that at this stage," it says, "it is rumour with no independent verification".
The Palestinian Authority said it, too, had no information confirming the fate of Alan Johnston.
Little known group
Senior Palestinian official Saeb Erekat repeated a call for those holding Mr Johnston to free him.
There have been numerous appeals for Mr Johnston's safe release
"Such despicable acts of abducting foreign journalists and others continue, the only (thing) that this is doing is destroying us as Palestinians, destroying the just cause of the Palestinian people," he told reporters.
"So I urge those who abducted Johnston, instead of circulating rumours, and I hope these rumours are only rumours, is to release him immediately and without any conditions."
A spokeswoman for the Foreign and Commonwealth Office in London said they were also aware of the reports and were urgently looking into them.
The group claiming to have killed Mr Johnston is little known in Palestinian areas, the BBC's diplomatic correspondent James Robbins says.
Its name means Brigades of Unification and Struggle.
Since Mr Johnston's abduction, there have been many calls by both governments and individuals for his safe release.
Palestinian journalists staged several strikes in protest at his abduction and rallies were organised by Mr Johnston's BBC colleagues.
The BBC held a day of action on Thursday with al-Jazeera, CNN and Sky News joining a special live programme.
The unprecedented broadcast paid tribute to Alan Johnston's work and detailed the dangers for correspondents working in the Gaza Strip.
An online petition calling for Mr Johnston's release has gathered more than 30,000 signatures.
Alan Johnston's father, Graham, in a second broadcast appeal for his son to be freed immediately, told his kidnappers: "Please think about what this is doing to my family".
Kidnappers have abducted dozens of foreigners in Gaza, but none have been held for as long as Mr Johnston.
The 44-year-old, who is originally 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.