A video of BBC reporter Alan Johnston, kidnapped in Gaza City on March 12th, has been posted on the internet.
The Army of Islam group posted the video of Mr Johnston (45) saying: "First of all, my captors have treated me very well.
"They have fed me well, there has been no violence towards me at all and I'm in good health."
In the video posted on an Islamic militant website Mr Johnston called for the lifting of international sanctions against the Palestinian government.
Looking fit and calm, the reporter spoke of the suffering Gazans had endured. He was shown from the waist up, wearing what looked like a red sweatshirt.
There date of the recording could not be established.
The Foreign Office said it was aware of media reports and was investigating the claims. The BBC said it was also investigating the reports, but had not yet obtained a copy of the video.
Palestinian officials have said they knew where to find Mr Johnston, but have held back on raiding the hideout at Britain's request, for fear of harming him.
The Palestinian government has instead tried to negotiate with the captors.
"We are aware of the reports and we are investigating urgently," said Simon Wilson, the BBC bureau chief in Jerusalem.
The company planned to issue a statement later today, a BBC spokesman in London said.
Last week, an aide to Palestinian Prime Minister Ismail Haniyeh said Mr Johnston was alive and well, and could soon be released.
He has been missing far longer than any foreigner kidnapped in Gaza, and his disappearance has touched off numerous protests and solidarity marches in London and the Palestinian territories.
AP