Kuwait's Al-Rai television quoted sources close to Carroll's captors on Friday saying they had "set a final deadline of 26 February to have their demands met, or implement the punishment stipulated by [their] religious law", which is death.
The US said it was making "every effort" to secure the release of the 28-year-old journalist, who was freelancing for the Christian Science Monitor when she was abducted on 7 January in Baghdad.
Carroll had appealed for her release in a video that was broadcast on Thursday on the same private Kuwaiti channel, but Friday's statement did not feature any new footage of thee journalist or her captors.
She is being held by a group calling itself the Brigades of Vengeance, which has called for the freeing of all female prisoners from Iraqi jails as a condition for her release.
The chairman of al-Rai television, Jasim Budai, said that information made available to him revealed that Carroll was being held in a house in central Baghdad.
'In good health'
"Sources related to the captors... asserted to us that the journalist (Carroll) is in a safe house in the centre of Baghdad," he said.
"She (is staying) with the wives of several of the kidnappers and in good health. She is performing domestic work with these women," he added, quoting the sources.
Budai said al-Rai channel had received a letter from the captors with Thursday's video which was handed to the Kuwaiti authorities, without divulging its content.
"She (is staying) with the wives of several of the kidnappers and in good health"
Jasim Budai ,
"There is one demand in this letter," he said, refusing to disclose it "for the sake of the journalist's safety."
He said the sources denied also that the captors were responsible for the killing of the local interpreter who was with Carroll.
In Washington, the State Department spokesman Sean McCormack said: "We continue to make every effort to secure her release."
He said the US government would not disclose any detail of its efforts to secure Carroll's release, explaining that "we certainly don't want to do anything that would in any way negatively affect the chances of her being returned safe and sound."