With the embattled 69-year-old dictator uncharacteristically silent, loyalist troops battled the rebel forces with truck-mounted machine guns, tanks and snipers, The New York Times reported.
Gadhafi's whereabouts were unknown to the rebels and the outside world.
"We do not know if he is inside or outside Libya," Mustafa Abdel-Jalil, chairman of the rebels' National Transitional Council, said at a news conference in Benghazi. The area around Gadhafi's compound in Tripoli remained under loyalists' control, he said.
The Times said foreign journalists who made their way through some Tripoli neighborhoods Monday afternoon said it was too soon to say the rebels had wrested control of the city from the Gadhafi forces. One rebel fighter told al-Jazeera the insurgents held 80-85 percent of the city.
"A great majority of the capital of Tripoli is under freedom fighters' control," said Guma el Gamaty, the Britain-based coordinator for the Transitional National Council.
International recognition of the legitimacy of the rebels' authority to govern expanded with Egypt calling them "the new regime" and saying the council would take over the Libyan Embassy in Cairo and assume Libya's seat on the Arab League, the Times said.
Gadhafi's regime still controlled some facilities, including a hospital, military barracks and the Rixos hotel, where international journalists are staying, CNN reported.
The Libyan leader's forces near the hospital location were negotiating their surrender, Jumma Ibrahim, a rebel military spokesman, said.
Celebrations in Tripoli's Green Square, which rebels renamed Martyrs' Square, were diminished after they learned Gadhafi's forces were heading to the gathering place, CNN said.
Gunfire and explosions were heard coming from the direction of Gadhafi's Bab al-Aziziya compound Monday.
Much of the compound has been destroyed by NATO airstrikes, operating under a U.N. Security Council resolution authorizing the use of force to protect Libyan civilians. Since NATO began the operation in March more than 7,400 strike sorties in Libya have been conducted.
In Zawiyah, residents celebrated the news of the apparent imminent toppling of the Gadhafi regime by firing guns in the air, shooting fireworks and chanting, "Libya is free!"
In Benghazi, the de facto rebel headquarters, one resident called the gains as joyous as "New Year's Day."
Gamaty said opposition leaders believed Gadhafi was hiding in Tripoli or fled to southen Libya, Chad or Algeria.
"Those are the only two neighboring countries that have been showing support for him," Gamaty said.
Gamaty told CNN setting up the political framework for a new Libyan government could take 18-20 months.
Libyans seeking an end to Gadhafi's four-decade rule cheered Sunday on news that two of the ruler's sons, Saif and Saadi, were arrested. On Monday, Mohammed Gadhafi, a third son, was taken into custody.
The International Criminal Court in The Hague, Netherlands, issued a warrant for Saif, alleging crimes against humanity. Arrest warrants also were issued for the elder Gadhafi and his intelligence chief, Abdullah al Sanussi.
CNN reported ICC Chief Prosecutor Luis Moreno-Ocampo indicated the court planned to begin negotiating for Saif Gadhafi's transfer to the Netherlands.
Mahmud Nacua, the Transitional National Council representative in London, said the council will move to Tripoli from Benghazi and establish a temporary government soon, The Wall Street Journal reported.
"There is a plan. ...There will be no [power] vacuum," Nacua said. "[The] NTC will move soon from Benghazi to Tripoli and they will appoint a new transitional government which will rule the country."
British Prime Minister David Cameron, who returned from a vacation early for a meeting of the National Security Council, said it was clear "that the end is near for Gadhafi," the BBC reported.
The Libyan leader had "committed appalling crimes against the people of Libya and he must go now to avoid any further suffering for his own people," Cameron said.
U.S. President Barack Obama said in a statement Sunday Gadhafi and his inner circle had "to recognize that their rule has come to an end" and called on Gadhafi "to relinquish power once and for all."
He called on the council to pursue "a transition to democracy that is just and inclusive for all of the people of Libya."
"The future of Libya is now in the hands of the Libyan people," Obama said, vowing the United States would work closely with the council and "continue to insist that the basic rights of the Libyan people are respected."
Earlier, Gadhafi issued a series of defiant audio statements calling on people to "save Tripoli" from the rebel offensive. He said Libyans were becoming "slaves of the imperialists."
"We will fight to the last drop of blood," he said in one statement. "We will never give up."
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