With US-led foreign forces nearing a complete withdrawal of troops, the Taliban have made swift territorial gains over the last two months but have not yet captured any provincial capitals.
"Since Thursday morning the Taliban have launched attacks from several directions on Lashkargah city," a government official told Reuters on the condition of anonymity. Lashkargah is the capital of Helmand, a southern province that borders Pakistan.
The official added that Afghan security forces have thus far held back the Taliban's attempt to take the city with the help of the Afghan air force, but operations were hampered by the presence of civilians in the area.
"Hundreds of families have left the area and moved to other safer places," Hafiz Ahmad, a resident of one Lashkargah neighborhood where clashes are ongoing, told Reuters.
He said those unable to move had locked themselves in their homes, and the city wore a deserted look as gun and artillery fire reverberated through neighborhoods.
A United Nations report this week said civilian casualties had been surging in recent weeks in Afghanistan, with as many killed in May and June as in the previous four months. The report did not cover casualties in July, when fighting has intensified further.
Abdul Majid Akhundzada, a Helmand provincial council member, said the Taliban had captured several areas of Lashkargah, and fighting was intense in an area close to the city's airport.
He expressed fear that the city could soon fall to the Taliban.
The United States' top military commander in the region has said the US air force has increased air strikes to support Afghan forces reeling from Taliban advances - but declined to say whether this would continue after their military mission ends on Aug. 31.
Meanwhile, some 200 Afghans were set to begin new lives in the United States on Friday as an airlift got under way for translators and others who risk Taliban retaliation because they worked for the United States during its 20-year war in Afghanistan, US officials said.