Libya’s UN-backed government says 21 people have been killed and 27 wounded in fighting near the capital, Tripoli.
US Secretary of State Mike Pompeo has called for an immediate halt to the fighting and called for talks.
Forces under Gen Khalifa Haftar have advanced from the east with the aim of taking Tripoli.
Prime Minister Fayez al-Serraj has accused him of attempting a coup and says his forces will not be allowed to continue.
Among the dead was a Red Crescent doctor killed on Saturday. Gen Haftar’s forces said they had lost 14 fighters.
Earlier the UN appealed for a two-hour truce so casualties and civilians could be evacuated, but fighting continued.
And in a statement, Secretary of State Pompeo said the US was “deeply concerned about fighting near Tripoli” and stressed the need for talks.
“This unilateral military campaign against Tripoli is endangering civilians and undermining prospects for a better future for all Libyans,” the statement said.
International powers have begun evacuating personnel from Libya amid the worsening security situation.
Libya has been torn by violence and political instability since long-time ruler Muammar Gaddafi was deposed and killed in 2011.
Gen Haftar’s Libyan National Army (LNA) forces have been carrying out a multi-pronged attack from the south and west of the city since Thursday.
The UN said its call for a humanitarian truce had been ignored and emergency services said they had not been able to enter the areas where fighting was taking place.
However a UN spokesman told AFP that they were “still hoping for a positive response”.
On Sunday the LNA said it had carried out its first air strike, a day after the UN-backed Government of National Accord (GNA) hit them with air strikes on Saturday.
Fighting has continued around the disused international airport south of the capital that Gen Haftar earlier said his forces had seized.
Forces loyal to the GNA have slowed the advance and on Sunday a GNA spokesman told Al-Jazeera TV that the GNA now intended to “cleanse” the whole of the country.
