International powers have begun to evacuate personnel from Libya in a deteriorating security situation.
The US Africa Command said it had mislaid a contingent of US troops, while India said it had evacuated peacekeepers to Tunisia.
Rebels among themselves Gen Khalifa Haftar has advanced from the East with the aim of conquering the capital Tripoli.
The UN-backed Prime Minister accuses him of attempting a coup and says rebels are being hit by force.
Libya was torn apart by violence and political instability since long-time ruler Muammar Gaddafi was deposed and killed in 201
- Why is Libya so lawless?
- Who is the military deputy Khalifa Haftar?
Who evacuated and why?
Gen Haftar started his offensive four days ago and the fight on the outskirts of the capital has led some international groups to respond to the deteriorating security situation.
US Africa Command, in charge of the US military operation s and Liaison in Africa, said that due to the "growing unrest", a contingent of US troops had been relocated temporarily, but no further details had been provided on the numbers.
There were reports of a fast amphibious vehicle being used in the operation.
Indian Foreign Minister Sushma Swaraj said his entire contingent of 15 peacekeepers from the Central Reserve Police Force had been evacuated from Tripoli because the "situation in Libya suddenly worsened".
Italian multinational oil and gas company Eni decided to evacuate all its Italian staff from the country.
The United Nations should also remove unnecessary personnel.
Residents of Tripoli have reportedly started stocking food and fuel. BBC Arab Affairs Editor Sebastian Usher says many of those near the fighting are staying in their homes for fear of looting should they leave.
Some fear a long operation that Gen Haftar undertook to capture the eastern city of Benghazi by Islamist fighters.
What happens to the fighting?
The battle continued on Sunday at the disused international airport south of the capital, of which Gen Haftar had previously claimed that his troops had been occupied.
Gen Haftar's Libyan National Army (LNA) forces have perpetrated a multiple assault from the south and west of the city.
Forces loyal to the UN-backed Government of the National Agreement (GNA) have now slowed the advance.
"Intensive air strikes" were carried out by GNA aircraft 50 kilometers south of Tripoli on Saturday.
A militia from Misrata told AFP that they had contacted the GNA and sent armed vehicles to Tajura in the eastern suburbs to counter the counterattacks LNA  9026] Who are the opposing forces?
Libya has been plagued by riots since the fall of Colonel Gaddafi. Dozens of militias operate in the country.
Recently, they allied themselves with either the UN-backed GNA based in Tripoli or the LNA of Gen Haftar, a harsh anti-Islamist backed by Egypt and the United Arab Emirates and heavily in eastern Libya.  Gen Haftar helped Col Gaddafi take power in 1969, before leaving with him and going into exile in the United States. He returned in 2011 after the uprising against Gaddafi began, and became a rebel commander.
The Government of the United Nations was formed during talks in 2015, but struggled to enforce national control.
Prime Minister Fayez al-Serraj provided a TV address On Saturday he said he would defend the capital.
Mr. Serraj said he had made concessions to Gen Haftar to avoid bloodshed, only to be "stabbed in the back".
Back to first place?
Analysis by Rana Jawad, correspondent of the BBC in North Africa, in Tunis
The resistance of the rogue general suggests that despite international condemnation of his recent actions, he believes he can only deal with military ones Secure a place in Libya's future political development.
Diplomats are worried because the type and timing of the attack makes it unlikely that he will kick back if he is not defeated.
Few thought he would start this operation, which he had long threatened, because he threatened for a long time. They believed that the ongoing talks that had taken him from Paris to Palermo and the United Arab Emirates for more as one year would gain time until a new political solution was found through negotiations and a possible electoral process.
Today, Western nations have few maps to de-escalate the violence and return to a position where they may need to start over.
Are peace talks planned?
UN-supported talks aimed at creating a roadmap for new elections UN envoy Ghassan Salame insisted that the talks be continued unless serious obstacles were prevented. He said, "We will not give up this political work quickly."  UN Secretary-General Antonio Guterres was in Tripoli last Thursday to discuss the situation.
But Gen Haftar said his troops will not stop until they have defeated "terrorism".