The G7 group of major industrial nations has urged all parties "to immediately halt all military activity".
The operation, dubbed 'Volcano of Rage', is aimed to repel the approaching Libyan National Army (LNA), led by Field Marshal Khalifa Haftar, who vowed earlier this week to seize Tripoli and liberate it from "militias and terrorists", uniting the country.
Hifter then gave orders for his forces to march on Tripoli, even as UN Secretary General António Guterres was in the Libyan capital preparing for a much-anticipated reconciliation conference between the warring sides meant to pave the way for long-delayed elections.
The sudden offensive against Tripoli by Libyan strongman Khalifa Haftar has taken pro-government forces and the worldwide community by surprise.
United Nations troops in the city have been placed on high alert.
Libya has been torn by violence and political instability since long-time ruler Muammar Gaddafi was deposed and killed in 2011.
What's happening on the ground?
The Libyan air force, which is nominally under government control, targeted an area 50km south of the capital on Saturday morning.
Gen Haftar spoke to Mr. Guterres in Benghazi on Friday, and reportedly told him that his operation would not stop until his troops had defeated "terrorism".
Al-Mesmari said Hifter's forces declared Tripoli a no-fly zone for warplanes.
There are now reports troops have taken the capital's airport, which has been closed since 2014 - although these are disputed.
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".
Fayez Sarraj, chairman of the Presidential Council of Libya, said his government had offered concessions to Hifter "to avoid bloodshed and to end divisions" and was surprised by Hifter's order to take the capital.
He did not provide details on the number of USA troops that have been withdrawn or how many remain in the country.
Tripoli-based army spokesperson Gununu said that troops are advancing on "all frontlines" and have captured "many military vehicles" from Haftar's forces. "There can be no military solution to the conflict", he said.
Eastern Libyan troops have advanced into the southern outskirts of the capital Tripoli in a unsafe thrust against the internationally recognized government.
They were also pushed back from a key checkpoint west of the capital, less than 24 hours after seizing it during their lightning offensive towards Tripoli.
The forces allied to Haftar also said they took control of the villages of Qasr ben Ghashir, Wadi al Rabie and Suq al Khamis south of Tripoli, in a statement.
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