American B-52 bombers have been pounding mountains in the area relentlessly, while local militia move in with tanks and fighters.
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Commander Mohammed Amin said they believed Bin Laden was within range of their tanks, and said he had reliable reports that the Saudi-born dissident had been seen in the area four days ago.
On Sunday, US Vice-President Dick Cheney said the US expected whoever found Bin Laden and senior members of his al-Qaeda network to hand them over.
Mr Cheney said Washington wants to try Bin Laden and his ally, Taleban spiritual leader Mullah Mohammad Omar, in newly-authorised military courts - if they are captured alive.
"They're exactly the kind of people the tribunals were established for," he said, referring to the controversial military tribunals authorised by President George W Bush to try foreign terrorism suspects.
The UK, meanwhile, has moved swiftly to reassure Washington that if its troops catch Bin Laden, he will be handed over immediately.
Defence Secretary Geoff Hoon had said the European Convention on Human Rights meant the UK could not hand over the main suspect in the 11 September attacks on the US if he faced the death penalty.
"Mr Hoon was talking hypothetically about him arriving on UK shores, " a government spokesman in London said.
In other developments:
US officials said fighting on the ground between al-Qaeda fighters and anti-Taleban forces in the area was "very, very fierce".
But Mohammed Amin says his forces are making slow but steady progress up the mountains.
However, one fighter contacted by radio by the Associated Press insisted the numbers were greatly exaggerated, saying he was part of a group of only 84 Arab fighters.
"I swear by Allah that Osama is not present here," he told the news agency.
Mr Cheney has confirmed the existence of a videotape which allegedly shows Bin Laden talking about the 11 September attacks which sparked the current conflict.
International justice
The US vice-president said the video, apparently discovered by US forces in the eastern Afghan town of Jalalabad, showed Bin Laden displaying "significant knowledge of what happened" on 11 September.
Other US officials, quoted anonymously, told news agencies that the tape showed Bin Laden recalling how he tuned in to news coverage hours before the attacks. After the first plane struck the World Trade Center in New York, Bin Laden told those watching with him that more destruction was to come.
Asked if Bin Laden was hiding in one of the massive cave complexes in the Tora Bora region, Mr Cheney said: "I believe so".
He added: "The preponderance of the reporting at this point indicates that Mullah Omar is still down in the Kandahar region someplace and that Osama Bin Laden is also still in Afghanistan in the general area of Tora Bora."
Hamid Karzai, due to take over as Afghanistan's interim leader on 22 December, said he had no idea where Bin Laden was.
But he added: "He has killed thousands of our people. If we catch him he will be given to international justice."
From the outset of its military campaign, the US has made it clear that its primary objective is to hunt down Bin Laden and bring him to justice.
Possible escape routes from Afghanistan are being patrolled by helicopters, military vehicles and soldiers on foot.