Downing Street has stressed that the government's position on no negotiating with terrorists remains unchanged.
Ken Bigley was shown sobbing behind bars
Tony Blair said he would talk to the kidnappers holding Ken Bigley in Iraq but said it would be "completely wrong" to give in to their demands.
His comments followed the release of a video showing Mr Bigley caged, and again pleading for help.
On Thursday Mr Bigley's brother Philip said they would never give up hope.
"Tony Blair is lying. He doesn't care about me," Ken Bigley, 62, said in the video appeal shown on al-Jazeera television on Wednesday.
"My life is cheap. He doesn't care about me."
Looking unwell and distraught he called on Mr Blair to meet the demands of his kidnappers to free female prisoners in Iraq and said his captors did not want to kill him.
The video comes as dozens of people have been killed in bomb attacks on coalition troops in Baghdad, and US air strikes in Falluja have killed at least three people.
Al-Jazeera also claimed on Thursday that 10 more hostages have been seized in Iraq. None are believed to be British.
Iraq interim Prime Minister Iyad Allawi who is presently in London called for the kidnappers to free Mr Bigley.
"It is repugnant to take an innocent man such as Kenneth Bigley and to use him as a political pawn in this way," he said, in a speech to the International Institute of Strategic Studies.
Earlier Tony Blair said he felt "absolutely sickened" by the situation and said he would talk to the hostage takers if they contacted him.
He told the BBC: "If you take a situation here in this country, if a hostage is taken and the police turn up, they will talk to the people who are holding the hostage.
"They are not going to give in to the hostage takers' demands, that's a different matter all together, that would be completely wrong."
16 Sept: Two Americans and Ken Bigley taken hostage
18 Sept: All three appear in video with captors
20 Sept: American Eugene Armstrong beheaded
21 Sept: American Jack Hensley beheaded
22 Sept: Ken Bigley appears on video, appealing for help from Tony Blair
29 Sept: Second video plea made by Bigley
Mr Straw, who will meet members of the Bigley family on Thursday, said the government "had to explore every avenue that is open".
He told the BBC Breakfast programme: "There have been plenty of hostage-taking situations where, in the end, the hostages have been released without the payment of a ransom, without the release of prisoners."
Defending the government's decision not to negotiate with terrorists, he said: "If we do not have this position there would be many many more people who would be kidnapped and the world would be less safe."
On Thursday Paul Bigley welcomed Mr Blair's comments but said he had "political handcuffs" on.
"A terrorist is not going to pick up the phone and call
you, it is just not going to happen," he said.
Speaking on BBC News 24, another brother, Philip, said the family had been "thrown into this nightmare" but had no other option but to handle it.
"We are taking it minute by minute," he said.
Sombat Bigley has appealed for clemency for her husband
Mr Bigley's wife, Sombat, said she was "still hopeful".
"I renewed my hope in the last few days when they tied him and wanted to kill him within 24 hours, but they didn't," she said.
The family also emphasised the strain which the ordeal was placing on Mr Bigley's mother, who is in hospital being treated for stress.
Mr Bigley was taken hostage by the hardline Tawhid and Jihad group along with the two American colleagues.
He has appeared in videos before, including on 22 September when he pleaded for his life.
The two Americans hostages have been beheaded.