Two senior Taleban leaders have been captured in heavy fighting for the southern town of Musa Qala, the Afghan defence ministry has said.
The Taleban said they have fortified their positions in Musa Qala
Afghan and Nato forces are trying to recapture the town, the only major Afghan centre in Taleban hands.
Twelve Taleban fighters, two children and a British soldier are reported killed in the battle. A second Nato soldier died in the area on Sunday.
The Taleban still control the town but have been forced back, reports say.
A statement from the Afghan defence ministry identified the two local Taleban leaders as Mullah Mateen Akhond and Mullah Rahim Akhond.
A defence ministry spokesman said the battle was going well so far.
"The reports which we have received from the site so far indicate that most of the enemy personnel have laid down their weapons and are leaving the area in civilian clothes," said the spokesman, Gen Mohammad Zahir Azimi.
The Defence Ministry's claims could not be independently confirmed.
The joint force of Afghan, British and US troops is reported to have Musa Qala surrounded from all sides and is edging closer to the town itself.
The assault was launched late on Friday and there have been fierce fire fights and heavy bombardments since then.
Afghan troops are playing a key part in the operation
Exchanges of gunfire on Sunday morning were later reported to have died down.
A British military spokesman said Nato forces were helping to pave the way for Afghan troops to seize and hold the town.
"Once the door is kicked in, the Afghan army will enter," said Lt Col Richard Eaton.
The Afghan defence ministry called on the Taleban fighters to lay down their weapons "or face waves of attacks".
The Taleban have held the town since February, when they retook it after British troops pulled out last year.
It is the only major town held by the Taleban and is at the centre of a major opium poppy growing area.
The heavy blows from the ground and the air seem to have forced the Taleban to pull back closer to the centre of Musa Qala, but they say they withdrew from two frontline villages because of civilian casualties there.
Nonetheless, Taleban commanders have said they will defend the town from fortified positions covered by minefields.
Musa Qala is believed to be mostly empty of civilians, many of whom fled after tribal elders were told of the attack and warning leaflets were dropped from the air, says the BBC's David Loyn in Kabul.
UK Defence Minister Des Browne, who is in Kabul, said the town had taken on iconic importance.
The Taleban took it over in February, in contravention of a controversial deal brokered with tribal elders when British troops withdrew.
It has since become the main centre of drugs trading in Afghanistan, our correspondent says.
The assault is the first major operation where the new Afghan army is playing a leading role.