People close to the kidnappers told the private TV channel earlier Friday that Carroll is "in a safe house owned by one of the kidnappers in downtown Baghdad with a group of women," Jassem Boudai told The Associated Press.
Carroll, who was kidnapped in Baghdad on Jan. 7, appeared in a video broadcast Thursday by Al-Rai in which she pleaded with the authorities to do whatever her abductors demanded, saying "there is a very short time."
"Please just do whatever they want, give them whatever they want as quickly as possible," she said, adding she was speaking on Feb. 2.
Her kidnappers, who call themselves the Revenge Brigades, have demanded the release of all Iraqi women held in U.S. military and Iraqi jails.
Boudai said Al-Rai had received a message from "sources close to the kidnappers." He said the message was not conveyed in a videotape, but "another method." He declined to how it was received and whether the message was delivered to Al-Rai's head office in Kuwait or its bureau in Baghdad — where Thursday's video of Carroll was received.
"Sources close to the kidnappers informed Al-Rai TV that the kidnappers have set a Feb. 26 deadline for their demands to be met or they will execute her according to Shariah," he said, referring to Islamic law.
He said the kidnappers have "more specific demands than releasing all women from jail," but he refused to disclose these demands.
He said the sources told Al-Rai that Carroll was in good psychological condition and was doing housework with the other women in the place where she was being held.
According to the sources, the kidnappers denied they killed Carroll's translator when the abducted her at gunpoint, as has previously been reported.
Carroll is a freelance reporter for The Christian Science Monitor.
Late last month, the U.S. military freed five Iraqi women detainees, but American officials insisted the release was not linked to the demand by Carroll's abductors.
The U.S. military was believed be holding about six more. It was unclear how many women were held by Iraqi authorities.
Some 250 foreigners have been taken captive since the 2003 U.S.-led invasion that toppled Saddam Hussein, and at least 39 have been killed.