Japan is refusing to withdraw troops from Iraq despite the death threats hanging over three of its citizens.
Picture of the frightened hostages are splashed across Japan's papers
The families of the three Japanese have pleaded with their government to give in to the demands of the Iraqi kidnappers and pull out of Iraq.
The Iraqi insurgents have threatened to burn the hostages alive unless Japan withdraws its forces within three days.
But Japan's Prime Minister Junichiro Koizumi said: "We should not give in to these despicable threats."
Japan has been stunned by video footage showing the three hostages blindfolded, and with a gun to their heads.
They are: Noriaki Imai, 18, who had planned to research the effects of depleted uranium weapons; 32-year-old photojournalist Soichiro Koriyama, and aid worker Nahoko Takato, 34.
The most harrowing footage, showing the hostages with knives to their throats and conveying the screams of the female hostage, Nahoko Takato, was not aired on Japanese television.
The hostages' families have met ministers to plead for the government to withdraw its troops.
"I can't bear the thought that my child might be burned alive," the mother of Soichiro Koriyama told a news conference, according to Kyodo news agency.
"Time is running out for my sister," said Ayako Inoue, Takato's younger sister. "As a family member, I can't just sit around watching television."
There is a presumed deadline of 2100 Japan time on Sunday (1200 GMT), but Mr Koizumi is under intense outside pressure not to give in.
US Vice President Dick Cheney arrives in Tokyo on Saturday at the beginning of an Asian tour, and is expected to urge him to stand firm in Iraq.
Japan's top spokesman, Yasuo Fukuda, said the demand to end the country's humanitarian mission, was not under consideration.
"That would be doing just what the terrorists want," he said. "We can't be beaten by them."
Mr Koizumi called an emergency meeting of his Cabinet, and has sent a senior Foreign Ministry official to Jordan to co-ordinate the rescue efforts.
The government had said it is yet to make contact with the group.
The BBC's Jonathan Head in Tokyo says that in safe and orderly Japan, the kidnappings, by a group calling itself the Mujahideen Brigades, are a source of nationwide anguish.
People in Tokyo react to the hostage crisis
The government cannot be sure how long public opinion in Japan can withstand the agony of a prolonged hostage crisis, our correspondent adds.
It adds up to the toughest test Mr Koizumi has faced. He took a huge political gamble when he made the decision to send 550 troops to Samawah, southern Iraq, earlier this year.
Hundreds of protesters gathered near parliament on Friday, shouting: "Prime minister, don't let the three be killed."
'Image at stake'
The Japanese public is sharply divided over the mission. Critics argue it violates the country's pacifist constitution, especially if the troops are drawn into violence.
JAPAN'S IRAQ MISSION
550 troops in Samawah
Helping to rebuild infrastructure
First time pacifist Japan has sent troops to combat zone
That possibility was underscored on Thursday, when mortars exploded near the troops' base.
But Mr Koizumi would jeopardise Japan's international standing if he pulled out, analysts say.
"Japan's image as a nation state is at stake," said Matake Kamiya, professor of security at the National Defence Academy.
South Korea, meanwhile, warned its citizens not to go to Iraq, after the abduction of seven South Korean missionaries, who were later released unharmed.
But the government in Seoul said it would go ahead with plans to send 3,600 troops to Iraq.