US and Afghan forces searching for militants on the Pakistani border have inflicted heavy casualties on al-Qaeda and Taleban fighters, officials say.
The Tora Bora mountains are remote and inhospitable
The governor of Ningrahar province said that the operation in Tora Bora was going "very well" and that al-Qaeda and the Taleban had suffered heavy losses.
But a spokesman for the governor, Nooar Agha Zwak, said he could not give precise casualty details.
Officials in Ningrahar say the area has many Afghan and foreign militants.
Local residents say that so far around 250 families have fled the area after two days of fighting, in which US and Afghan air and ground forces pounded al-Qaeda militants.
The US military said that "al-Qaeda and other violent extremist fighters" had been engaged in the fighting during a "combined arms assault using precision munitions".
Taleban and al-Qaeda fighters escaped the 2001 Tora Bora assault
There were no substantiated reports of civilian casualties in the fighting, a statement said, contradicting earlier reports that many civilians had been killed.
"The targets were carefully chosen to pinpoint enemy positions and eliminate the likelihood of harming innocent civilians," said the statement from US army spokeswoman Vanessa Bowman.
"This region has provided an ideal environment to conceal enemy support bases and training sites, as well as plan and launch attacks aimed at terrorising innocent civilians, both inside and outside the region.
"Credible intelligence indicated that the enemy fighters were gathered in this mountainous area in dug-in fighting positions."
In an interview with the AFP news agency, Capt Bowman said that the object of the offensive was to disrupt al-Qaeda and other militants who were massing in the region.
Pakistan is meanwhile reported to have deployed troops on its side of the border to prevent militants escaping from the US-led offensive.
"It has been done over the past three days and it was done in co-ordination with allied forces in Afghanistan," a security official told the Reuters news agency.
The Pakistani army says that at least 10 militants and four soldiers have been killed close to the Afghan border in separate incidents in North and South Waziristan.
Correspondents say it is unclear whether this fighting is in any way linked to events in Tora Bora.
Tora Bora was the scene of a failed major US operation to capture Bin Laden in 2001.
The region consists of a complex of caves, and is known as the last stronghold of al-Qaeda in Afghanistan.
Bin Laden and fighters from Arab countries are reputed to have used the caves in the 1980s during the resistance to the Soviet occupation of Afghanistan.
Bin Laden was last seen in the Tora Bora area in December 2001 - three months after he ordered the 9/11 attacks in New York - when US and Afghan forces tried but failed to capture him.
Correspondents say that it is widely believed that he escaped into the northern tribal areas of Pakistan where he is still believed to be hiding - taking advantage of the porous border between the two countries.
News of the US offensive comes as three German nationals were killed, and a fourth wounded, by a roadside bomb, near the Afghan capital, Kabul.
They were travelling in a diplomatic convoy. Local police say the blast was caused by a remote-controlled bomb, which completely destroyed one vehicle.
In a separate incident, a British national working for a private security firm which guards the British embassy, was killed by unknown assailants in Kabul.