Radmanish said the governor's office, the police headquarters, the National Directorate for Security, the university and the airport remain under government control and that Afghan officials would quickly expel the Taliban from Farah.
Special forces from Kandahar and Herat were rushed to the city to take part in the fighting, which residents said began around midnight.
Mujahid said Taliban fighters launched a multipronged attack and overran a large number of checkpoints belonging to Afghan security forces in the city.
The U.S. -led military coalition in Afghanistan said in a tweet that it was providing air support to Afghan forces around Farah.
Afghan and North Atlantic Treaty Organisation warplanes also took part in the battle.
"The independent media is a cornerstone of democracy", Jim Mattis said, adding that the United States was "committed to defeating" the Islamic State group, which claimed responsibility for the Kabul attack.
Farah province, which borders Iran, has been the scene of intense fighting in recent years, and there have always been fears that the city is vulnerable.
Clashes between militants and security forces forced Farah residents to close shops and businesses.
The noise has "filled the city", said one resident who gave his name as Bilal, adding that he could see smoke rising from the direction of a building housing the NDS.More news: Indonesian police kill one, arrest several in Surabaya raids
Early Tuesday, Ahmad Zaki, a doctor at the city's central hospital, said that so far two dead police officers and 16 wounded security forces members had been brought to the medical facility since the start of the latest assault.
Fighters are targeting journalists in Afghanistan because they are weakened and want more news coverage in order to undermine the country's electoral process ahead of an expected vote in October, Pentagon chief Jim Mattis has said.
Mohammad Sarwar Osmani, a lawmaker from Farah, said residents were trying to get their lives back to normal.
"They kill one, and they claim three", Radmanish said.
Farah is a vast and sparsely populated province with tens of thousands of dollars generated under custom duty.
The assault comes as the Taliban are stepping up their spring offensive, in an apparent rejection of a peace talks overture by the Afghan government.
Farah, which borders Iran, has been the scene of intense fighting in recent years. In 2017, insurgents tried three times to overrun the capital, according to the Afghanistan Analysts Network.
Although the insurgents have been unable to take and hold any provincial centre, they are active across Afghanistan and the government has firm control over no more than 56 percent of the country, according to US estimates.
Farah is home to about 50,000 people and would be the second city to fall to the Taliban since the city of Kunduz fell twice in 2015 and 2016.