Bin Laden may be hiding in the Tora Bora complex
The fighters set out from the town of Jalalabad intent on reaching the area of Tora Bora, where Bin Laden - accused of masterminding the 11 September attacks on the United States - is thought to be in hiding.
One of the leaders of the anti-Taleban advance on Tora Bora said he was assembling a 3,000-strong force to hunt for the al-Qaeda leader.
Hazrat Ali also told the Associated Press news agency that a patrol comprising about a dozen of his men exchanged fire with al-Qaeda fighters in the Tora Bora area on Tuesday.
The al-Qaeda fighters abandoned a tank and fled to higher ground, he said.
Mr Ali said that weeks of US bombing had driven up to 1,200 al-Qaeda fighters high up into the mountains, where they were split into small groups.
At least 12 al-Qaeda fighters had been killed in the past two days, he added.
Opposition 'pushed back'
Meanwhile, in the south of the country, opposition forces reportedly withdrew from an area near the Taleban stronghold of Kandahar after meeting fierce resistance - apparently from foreign al-Qaeda fighters.
Ahmad Karzai, the brother of Pashtun leader Hamid Karzai, told Reuters news agency that there had been heavy clashes between Pashtun tribesmen loyal to former Kandahar governor Gul Agha and Taleban troops at Kandahar airport.
"There was very heavy fighting. Gul Agha's men tried to enter the airport itself, but they could not," he said.
"They are now back where they were, on the banks of the river about one or two kilometres from the centre of the airport."
A spokesman for Gul Agha's men said they had withdrawn so that the US could continue to bomb the area.
US resolve
At the latest Pentagon briefing, US Defence Secretary Donald Rumsfeld said two main groups of anti-Taleban fighters were closing in on Kandahar.
"The situation in Kandahar is complicated," he told reporters. "It's not easy, but one thing is clear: The Taleban and al-Qaeda will be driven from Kandahar."
Mr Rumsfeld said that although there was still hope that the Taleban might surrender control of the city, the US was prepared to use force if necessary to obtain it.
He added that the Taleban would have little chance of reinforcing their troops as all roads in and out of the city were under constant US surveillance.
Pentagon officials confirmed that a member of US special forces was shot and injured near Kandahar on Tuesday. His condition is reported to be stable.
The US is attempting to downplay claims of civilian deaths caused by American bombing of the area, saying there is little accurate information coming out of Afghanistan.
Mr Rumsfeld said it was "next to impossible" to obtain accurate figures of civilian casualties as the Taleban "lie and distort".
"We did not start this war, so responsibility for every single casualty in this war rests at the feet of al-Qaeda and the Taleban," he said.
He also accused the Taleban of using innocent civilians as targets, saying:. "The Taleban are in effect using the civilian population of Kandahar as human shields."