Sangin is a key district in Afghanistan's southern Helmand province.
The Taliban took over the entire district except for the police chief's compound and another compound, where a battalion of the Afghan National Army is based, a police official said.
He said that officers were running out of ammunition and food after several days of holding off the Taliban, but no help had reached them yet.
Afghan officials have not responded to repeated calls from CNN to explain the situation.
Afghan CEO Abdullah Abdullah said Monday that the government was working to relieve the forces and repel the Taliban. Meanwhile Afghan President Ashraf Ghani was in transit on a previously scheduled trip to Azerbaijan.
British troops deployed to Helmand
A small contingent of British troops was sent to Helmand over the weekend to provide support to embattled Afghan forces in the province.
The troops, the British Ministry of Defence said in a statement, were "part of the UK's ongoing contribution to NATO's Resolute Support Mission
," the training, advisory, assistance and counterterror mission in Afghanistan. They were deployed to Camp Shorabak, on the site of Camp Bastion, the former British Army headquarters in Afghanistan, the ministry said.
"These personnel are part of a larger NATO team, which is providing advice to the Afghan National Army. They are not deployed in a combat role and will not deploy outside the camp," the statement said.
December marks one year since NATO handed over security operations to the Afghans. Before that, British and American forces struggled for years to hold on to Sangin.
Stuart Gordon, a Helmand expert at the Chatham House think tank, told Britain's Press Association news agency that Sangin held a special significance to the British as more than 100 British troops had been killed there.
"Sangin became fairly totemic for the British because of the number of soldiers lost," he said.
A fertile region that is a key location in Afghanistan's poppy trade, Sangin lies in the south of the country in an area that has traditionally been a Taliban heartland.
"It was significant because of the routes it controlled and it was a very significant part of the resourcing of the political economy of Helmand, because it is a major center of drugs processing and drugs shipping," said Gordon.
It was strategically important because it linked Lashkar Gah, the Helmand capital, to districts in the north, he said.
If the Taliban gained control of Sangin, they would control supply routes to the districts, and valuable influence over neighboring provinces, he said.
"If Sangin falls, much of the north of Helmand is very much under Taliban control," he said.
"This is probably the worst of the scenarios that the British had in 2013 and 2014."
Mohammad Jan Rasolyaar, deputy governor of Helmand province, took an unusual step over the weekend by posting an open letter to Ghani on Facebook asking for help.
He warned that all of Helmand could fall to the Taliban if the President didn't take action.
Rasolyaar mentioned Sangin district in his message, saying its main bazaar and the government office were under heavy attack by the Taliban. During the recent intense fighting in Sangin and Greshk districts, 90 Afghan security forces had been killed, he said.
Omar Hamid, head of Asia-Pacific country risk at IHS, told CNN that in Sangin, and before that in Kunduz
, the Afghan government forces struggled to put up a fight against the Taliban without foreign backup.
"The problem is where the Afghan forces have to fend for themselves," he said.
"They're fine as long as they're being assisted and they're being provided air cover and things like that by Western forces. But it's when they're left to their own devices that they seem to struggle."