The US military said in a statement that it had A-10 warplanes over the city, "which remains under government control".
Officials said the insurgents were close to overrunning four police districts and the city's headquarters of the National Directorate of Security, the main intelligence service, was under heavy attack.
Fared Bakhtawer, head of the provincial council, offered a different picture, however, saying that several security checkpoints in Farah were overrun by the Taliban.
Afghan forces have been struggling to hold back the resurgent militant group since the withdrawal of North Atlantic Treaty Organisation combat forces at the end of 2014.
The Ministry of Defense (MoD) claims that the Taliban militants have suffered heavy casualties during the attack on Farah city.
Heavy fighting has broken out in the western Afghan city of Farah after the Taliban launched a major offensive overnight to capture the provincial capital.
Taliban fighters stand guard as senior leader of a breakaway faction of the Taliban Mullah Abdul Manan Niazi, not pictured, delivers a speech to his fighters, in Shindand district of Herat province, Afghanistan, May 27, 2016.
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. He said casualties were high among security forces, but couldn't provide a precise number.
Salangi said more than 300 Taliban fighters were killed in the gunbattle.
Ashraf Ghani Afghanistan's president speaking at an event in Herat in February
The US carried out drone strikes overnight and the Afghan army is still clearing the city, according to Afghan and North Atlantic Treaty Organisation officials.
Mohammad Sarwar Osmani, a lawmaker form Farah province, also confirmed the Taliban attacks.
Gen. Mohammed Radmanish, chief spokesman for the Defense Ministry, said the situation "was under control".
The insurgents released a statement warning residents to stay inside their homes and "stay calm".
Many radio and television channels in the province have stopped broadcasting, fearing for their employees' lives, according to media watchdog Nai.
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 is in dire need of air support", Dadullah Qane, a provincial council member, said by telephone as the fighting went on in the morning.
It's also an important economic area with the multi-billion-dollar TAPI (Turkmenistan, Afghanistan, Pakistan and India) gas pipeline planned for the region.
Farah has been the scene of intense fighting in recent years. Officials have warned of a deteriorating security situation as insurgents have entered the capital several times.
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 USA estimates.
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