Residents in Tripoli are reported to be staying indoors as pro-government forces and the rebels both claim to control most of Libya's capital.
The rebels were met by jubilant crowds when they stormed into Tripoli at the weekend, but by Monday forces loyal to long-standing leader Muammar Gaddafi began fighting back.
The whereabouts of 69-year-old Colonel Gaddafi is unknown and he is believed to be in hiding in a large compound near Bab al-Aziziyah where explosions have been heard on Tuesday.
Bab al-Aziziyah has been the focal point of fighting in the capital and rebel leaders have said that they do not expect the huge complex to fall easily.
NATO officials are reportedly concerned that Colonel Gaddafi may have stockpiles of mustard gas. Defectors from his regime have told NATO that Colonel Gaddafi has tonnes of the poisonous gas which he may use against the rebels.
Meanwhile, one of Colonel Gaddafi's sons, Saif al-Islam Gaddafi, has resurfaced a day after the rebels claimed he was in custody. He has talked to journalists and was seen on television being besieged by supporters.
The BBC reports the 38-year-old turned up at the five-star Hotel Rixos early on Tuesday. He said he did not care about an arrest warrant issued against him by the International Criminal Court in The Hague on charges of crimes against humanity.
Asked if Colonel Gaddafi was safe and in Tripoli, Saif al-Islam replied: "Of course."
In a broadcast on Sunday before State TV went off the air, Colonel Gaddafi said he would stay in Tripoli until the end. If captured alive, he faces war crimes charges before the International Criminal Court.
Meanwhile, members of the Libyan rebels' National Transitional Council say they plan to fly to Tripoli in the next day or so to start work on forming a new government.
A BBC correspondent in the city of Benghazi says there is optimism in their ranks that by the middle of the week Tripoli airport will be secure enough to allow them to move.
International calls to step down
World leaders have urged Muammar Gaddafi to step down.
United Nations Secretary-General Ban Ki-moon called on Colonel Gaddafi's forces to stop fighting immediately and make way for a transition of power.
Mr Ban said the UN will hold an urgent meeting this week with the Arab League, the African Union, the Organisation of Islamic States and the European Union to discuss the situation.
United States President Barack Obama has urged the rebels not to seek justice through violent reprisals.
In a statement, he said America will be a friend and partner to Libya, but that the future of the country is in the hands of its people.
Mr Obama praised the Libyan people for what he called their extraordinary sacrifice in defying the Gaddafi regime and fighting to overthrow it.
