The LNA said it has reached the southern outskirts and has taken the disused global airport, 24 kilometres from central Tripoli, but GNA military reportedly refuted this.
The rival militias, which are affiliated with a United Nations -backed government in Tripoli, said they had also carried out airstrikes, slowing Hifter's advance. The UN mission to Libya has called for a two-hour cease-fire on Sunday in parts of Tripoli to evacuate civilians and wounded people.
Libya has been gripped by unrest since the 2011 uprising that overthrew and killed longtime dictator Moammar Gadhafi.
G7 foreign ministers meeting in the French coastal city of Dinard on Saturday called on Haftar to halt his offensive on Tripoli.
Rebel forces are advancing on Tripoli in a multi-pronged attack from the south and west of the city, although they have reportedly been slowed by pro-government fighters. It is the first public mention that America had troops on Libyan soil.
Still, American forces and diplomatic staff have continued to work in Libya with a special focus on helping the UN-backed government maintain its rickety grip on portions of the country.
The US joined fellow members of the G7 economic group - Canada, France, Germany, Italy, Japan and the United Kingdom - in condemning the Tripoli fighting, saying: "We firmly believe that there is no military solution to the Libyan conflict". "I still hope it is possible to avoid a bloody confrontation in and around Tripoli", he said on Twitter.
"Of course we consider that the most important thing is that (military) operations there do not lead to bloodshed".
The head of Libya's internationally-recognised government, Fayez al Sarraj, on Saturday accused his rival Khalifa Haftar of "betraying" him after the military strongman launched an offensive against the capital Tripoli.
Hifter announced Thursday he was deploying his forces toward Tripoli, sparking fears that the tensions could be escalating out of control as militias from the western cities of Zawiya and Misarata said that they have mobilized to confront Hifter.
The UN has declared that the gathering will go on as planned.
Haftar is "motivated by personal interests and his own delusions" and is trying to "plunge the country into a cycle of violence", said Sarraj. The UN Security council has issued a similar call. Last week, Haftar-who enjoys varying degrees support from the UAE, Egpyt, Saudi Arabia, and Russia-ordered his militias to take the UN-backed Libyan government's seat of power in what could be a far more decisive military operation than most expected it to be.