But the commander of U.S. Central Command, General Joseph Votel, said on May 11 that the Taliban "cannot win militarily".
In a separate attack in Farah city, militants killed 11 police officers and seized a large quantity of weapons and equipment.
On Monday, CEO of the National Unity Government (NUG) Abdullah Abdullah said soldiers have been under siege for months, but relevant security officials are busy with other issues.
A Farah government official and several residents said security forces were engaged in police districts two and three, around four kilometres from the city centre.
The Taliban overran several security checkpoints in the city near Afghanistan's border with Iran. He said casualties were high but couldn't provide a precise number.
Samihullah Samim, a legislator from Farah, said parts of the city are under Taliban control, with heavy fighting just 300 metres from the governor's residence.
One of the attacks was claimed by the Islamic State of Iraq and the Levant (ISIL, also known as ISIS) group and the other by the Taliban.
The Taliban launched a coordinated assault on Farah shortly after midnight, and quickly overran checkpoints at the city's edge.
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Local military experts believed the Taliban offensive followed their loss of control over the southern province of Helmand.
The Afghan Defense Ministry said in a statement Tuesday that Afghan security forces were able to repel their attacks.
There was no immediate confirmation from NATO's mission in Kabul.
At least 10 people were killed and another 40 wounded as a result of a Taliban assault in the capital of eastern Nangarhar province on Sunday, an official said.
"If the security forces can't take control of the whole city by the end of the day, then it will be very hard to take control of the city anytime soon", Samim said.
Inside the city residents reported clashes were continuing.
"NDS forces in their headquarters are engaged in heavy clashes with the Taliban".
Another provincial council member, Dadullah Qani, confirmed Hissaini's comments.
"The sounds of explosions and gunfire have filled the city", one resident who gave his name as Bilal said, adding that he could see smoke rising from the direction of a building housing the NDS. Local residents fear the whole province may fall to the insurgents.
Farah Province has been a key battleground for the Taliban for months. There are plans for a section of the multi-billion-dollar TAPI (Turkmenistan, Afghanistan, Pakistan and India) gas pipeline to traverse it, a project the Taliban have pledged to cooperate with.