The government denied the reports, The Daily Telegraph reported. But residents of the capital said they heard shooting and rebel leaders said the fight for the city was under way, the British newspaper said.
"The fighters in Tripoli are rising up in two places at the moment -- some are in the Tajoura neighborhood and the other is near the Matiga airport," Col. Fadlallah Haroun in Benghazi told the Arabic satellite channel al-Jazeera.
Gadhafi on an audiotape broadcast early Sunday called the reports lies, CNN reported. He agreed with the rebels, however, that the six-month-long civil war is close to an end.
"They have lost and used everything, and their last resort is their lying campaign," he said. "Now, this is the ending."
While Tripoli has been subjected to repeated NATO airstrikes, there have been no visible anti-government protests in months. Gadhafi himself has stayed largely out of sight.
Rebel leaders say that has changed swiftly. The New York Times said it had been able to interview Tripoli residents by telephone who confirmed there was fighting in the city.
The Times said there were reports of thousands of refugees fleeing Tripoli.
Earlier Saturday, rebel leaders said they had complete control of Zawiyah, a port and site of a major oil refinery, 30 miles west of Tripoli. They also declared victory in Zliten, 100 miles to the east, and last week seized Gharyan in the south.
Ali Gliwan, a spokesman for the rebel council in Misurata to the east of Tripoli, told The Guardian that Zliten, 100 miles from the capital, was now in the hands of "thwar" -- or revolutionary forces -- and fighters were now outside al-Khums. Al-Khums, 70 miles from Tripoli, is the last major road junction.
"Misurata thwar linked with the Zliten thwar," Gliwan told the Guardian. "They are now establishing control of the town."
Gadhafi, who has ruled Libya since 1969, spoke on national radio Friday night denying rebel advances and urged citizens to join the army in quelling the rebellion.
"The blood of the martyrs is fuel for the battlefield," he said.
Meanwhile, the prospect of Tripoli coming under siege had the International Organization for Migration scrambling for means for a mass evacuation of foreign migrants, the reports said. IOM spokeswoman Jemini Pandya told The Daily Telegraph logistics didn't favor land routes for an evacuation.