"If the agent government does not meet our demands within 48 hours we will behead them (the hostages)," the Ansar al-Jihad group said in a statement dated Wednesday and carried on a Muslim website.
Its authenticity could not be immediately verified.
A spokesman for the Iraqi interim government said on Wednesday that a first cousin of the prime minister, the cousin's wife and another family member were seized from their home in Baghdad on Tuesday morning.
"With the help of God in this holy month (Ramadan) a unit of Ansar al-Jihad kidnapped three relatives of head of the Iraqi agents, Allawi, may God burn him and slaughter him," the statement said.
It demanded the release of all Iraqi female and male prisoners and an end to the assault on the city of Falluja.
Allawi ordered the American-led assault on Falluja to crush anti-US resistance before Iraq tries to hold an election in late January.
Commenting on the situation, an Iraqi political analyst, Nizar al-Samarrai told Aljazeera.
"When such an incident reaches the house of the prime miniter, the highest political official in the country, this means there is a complete security disorder."
"The country cannot control the situation and protect officials and their families," said al-Samarrai.
The analyst believes that what is happening in many Iraqi cities, particularly Falluja, will kick-start a new round of violence in the country.
Al-Samarrai says the abductions have been carried out by those harmed by Allawi's decision to attack Falluja. They are holding him responsible for all the victims of the attack, he added.