Quoting sources close to the captors, the private channel said the captors had "set a final deadline of February 26 to have their demands for all female prisoners in Iraq to be released, otherwise the 28-year-old journalist would be executed.
Al-Rai chairman Jassim Boodai refused to reveal the source but said the information is "fresh", according to CNN.
The United States said it is making "every effort" to secure the release of the Ms Carroll, who was freelancing for the Christian Science Monitor when she was abducted on January 7 in Baghdad.
News of the deadline comes a day after Ms Carroll appeared in a second video tape, where she appealed for her release, shown on Al-Rai.
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.
Mr Boodai told AFP that information made available to him revealed that Ms Carroll was being held in a house in central Baghdad.
"Sources related to the captors ... asserted to us that the journalist 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.
Appearing composed and dressed in a headscarf in the recent video, she urged US authorities to agree to the demands of her captors "as quickly as possible".
"I'm fine, please just do whatever they want," she said.
"There is very short time."
The appearance contrasted with a harrowing video of the journalist, aired Al-Jazeera television, in which she had made an emotional plea for her release with tears streaming down her cheeks.
Mr Boodai said the channel received a letter from the captors with Thursday's video which was handed to the Kuwaiti authorities, without divulging its content.
"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 Ms Carroll.
In Washington, State Department spokesman Sean McCormack said: "We continue to make every effort to secure her release."
On Thursday, a spokesman for the coalition forces in Iraq said the US military would soon release a fresh batch of 450 detainees, but did not specify whether there would be any women prisoners among them.
Last week the military released 50 male detainees, while in late January, more than 400 were released, including five women.
The US and Iraqi authorities have repeatedly denied any link between the detainee releases and Carroll's abduction.