The rebels and NATO said that Gadhafi loyalists were negotiating the fate of Sirte, a heavily militarized city some 250 miles (400 kilometers) east of the capital, Tripoli.
Mustafa Abdel Jalil, the head of the rebels' National Transitional Council, said that negotiations with forces in Sirte would end Saturday after the Muslim holiday of Eid al-Fitr, when the rebels would "act decisively and militarily."
We can't wait more than that," he told reporters in the eastern city of Benghazi. "We seek and support any efforts to enter these places peacefully. At the end, it might be decided militarily. I hope it will not be the case."
Col. Roland Lavoie, a NATO spokesman, said it's possible Sirte might surrender without a fight.
"We have seen dialogues in several villages that were freed - I'm not saying with no hostilities, but with minimal hostilities," he said.
Lavoie said NATO would continue its mission as long as civilians in the country are under threat, although the area around the capital, Tripoli, is now "essentially free."
Lavoie appeared to struggle to explain how NATO strikes were protecting civilians at this stage in the conflict. Asked about NATO's assertion that it hit 22 armed vehicles near Sirte on Monday, he was unable to say how the vehicles were threatening civilians, or whether they were in motion or parked.
A top rebel official, meanwhile, said their forces were closing in on Gadhafi.
Ali Tarhouni, a minister in the National Transitional Council, told reporters Tuesday that "we have a good idea where he is. We don't have any doubt that we will catch him."
He gave no further details.
The rebels also demanded that Algeria return Gadhafi's wife and three of his children for trial after they fled, raising tensions between the neighboring countries.
Safiya Gadhafi, her daughter Aisha and sons Hannibal and Mohammed entered Algeria on Monday, while Gadhafi and several other sons remain at large. In Washington, the Obama administration said it had no indication that Gadhafi himself has left the country.
Algeria's Health Ministry said that Aisha Gadhafi gave birth to a girl on Tuesday. The official provided no other information, including on where she gave birth. The official was not authorized to be publicly named according to ministry rules.
Algerian news reports had said Aisha's pending childbirth was one reason for Algeria's decision to take the family in.
The departure of Gadhafi's family was one of the strongest signs yet that the longtime leader has lost his grip on the country. Algeria's decision to host members of the Gadhafi clan is an "aggressive act against the Libyan people's wish," said Mahmoud Shammam, information minister in the rebels' interim government.
Rebels also said another Gadhafi son, Khamis, was likely killed last week in a battle south of Tripoli.
"We are determined to arrest and try the whole Gadhafi family, including Gadhafi himself," Shammam said late Monday. "We'd like to see those people coming back to Libya."
Rebel leaders said they were not surprised to hear Algeria welcomed Gadhafi's family. Throughout Libya's six-month uprising, rebels have accused Algeria of providing Gadhafi with mercenaries to repress the revolt.
Gadhafi's children played important roles in the country's military and economic life. Hannibal headed the maritime transport company; Mohammed the national Olympic committee. Aisha, a lawyer, helped in the defense of toppled Iraqi dictator Saddam Hussein in the trial that led to his hanging.
David Nichols, a senior executive officer with Amnesty International, decried increasing calls in the international community for Gadhafi to be tried in Libya if he is caught.
Nichols told AP Television that Gadhafi and other officials who have been indicted for crimes against humanity should be sent to the International Criminal Court in The Hague, Netherlands because the new government in Libya will have neither the capacity nor the experience to try them fairly.
Rebels worry that if Gadhafi is not killed or captured, he will stoke more violence.
NATO reported hitting the 22 armed vehicles, three command and control sites, four radar installations and several other targets in the Sirte area Monday. Other targets were hit in contested regions south of Sirte.
Some 150 kilometers (90 miles) west of Sirte Tuesday, about a dozen armored, gun-mounted trucks were parked at a staging ground in barren desert. A highway overpass provided some shade for rebels, most dressed in T-shirts and camouflage pants.
Ismail Shallouf, a rebel commander at the staging ground, said patrols have gone 50 kilometers (30 miles) closer to Sirte, and occasionally exchanged fire with Gadhafi fighters.
"The leadership told us to wait for now," Shallouf said. "We don't have any information about the negotiations. Maybe there will be an assault after Eid," the holiday marking the end of Ramadan, the Muslim holy month of fasting.
Mohammed Amer, another local rebel commander, was dismissive of Gadhafi.
"Gadhafi has no more brigades left or equipment," Amer said. "He just has mercenaries who take his money."
A NATO officer, who asked not to be identified because of alliance rules, said on Monday there were clashes around Sirte, Bani Walid south of Misrata and Sebha further south.
In Tripoli, rebel leaders trying to set up a new government struggled with widespread shortages of water and fuel. In one neighborhood in the capital, dozens of motorists broke into a gas station Monday and filled plastic contains with fuel. Long lines formed at other gas stations.
Some residents filled containers with drinking water from large trucks, while others relied on wells. One of the water truck drivers, Ramzi Abu Shabaan, said the shortages were a small price to pay.
"I don't care if we go without water for two months even - frizz-head is gone - it's worth it," using a commonly used derogatory nickname for Gadhafi.
Shops selling clothes, shoes and toys opened for the first time since the rebels entered the city Aug. 20.
Children accompanied their mothers and fathers into shops to pick clothes and toys for this week's Muslim holiday.
"This will be the happiest Eid we celebrate," said Munira Omar, 30 who bought her two daughters hair clips and dresses.
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Laub reported from Tripoli. Don Melvin contributed from Brussels.
