Opposition Northern Alliance fighters, under the command of General Abdul Rashid Dostum, are now said to be within 30 kilometres (19 miles) of the strategic city of Mazar-i-Sharif.
The advances come as the United States and its allies continue a massive build up of armed forces in readiness for strikes against the Taleban.
President Bush has threatened strikes against the Taleban because of their refusal to hand over Osama Bin Laden, the prime suspect in the 11 September suicide plane attacks on New York and Washington.
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But the Taleban supreme leader, Mullah Mohammed Omar, on Monday appeared to dismiss the US threats.
"If America wants terrorism to end, it should withdraw its forces from the Gulf and end its partisanship in Palestine," the statement said.
The Taleban are saying thay no longer know where Bin Laden is.
But the Taleban defence minister says they have mobilised 300,000 "experienced" fighters for a holy war, or jihad against the United States.
The opposition advances in northern Afghanistan came after they captured Zari district in Balkh province.
A Taleban spokesman confirmed on Monday that they had lost control of Zari, the Afghan Islamic Press (AIP) reports.
The Taleban controls about 90% of Afghanistan.
Opposition Foreign Minister Abdullah Abdullah told reporters that if his forces managed to take Shulgar district in Balkh province, "the Taleban will be threatened in one of the major cities, Mazar-i-Sharif".
There has also been heavy mortar and artillery fire in the Panjshir Valley, some 70 kilometres (45 miles) north of the capital, Kabul.
Meanwhile, the political leader of the Northern Alliance, Burhanuddin Rabbani, arrived in the area bordering Tajikistan amid speculation he may hold talks with American officials.
Correspondents say the opposition is moving men and supplies into positions south of the border with Tajikistan in readiness for any new push into Taleban-held areas.
Under attack
AIP said Taleban forces were under attack in the northern provinces of Takhar and Samangan, east of Mazar-i-Sharif.
An opposition commander told Reuters news agency that Northern Alliance forces had also taken the district of Lawlash in the province of Faryah, where heavy fighting is continuing.
A local commander told Reuters news agency that the Taleban "are putting up tough resistance, but they are suffering lots of deaths".
The BBC's Sarah Nelson in Afghanistan says there are regular artillery exchanges but, according to local opposition tank commanders, the Taleban's return fire is diminishing.
They believe the Taleban are starting to withdraw in order to defend positions further south.
Foreign support
In a sign of ongoing diplomatic activity, the opposition's Foreign Minister Abdullah met the chief of Russia's general staff in Tajkistan on Sunday.
He said the opposition was also in close contact with American officials through its diplomats in the United States.
The Northern Alliance has said it has been in contact with the US State Department and is ready to help a US operation against the Taleban.
Correspondents say the opposition's detailed knowledge of the area and of Taleban military capabilities could prove invaluable to US forces in a strike against Osama Bin Laden and his associates.
In a statement, he said that Washington could not resolve the current crisis by killing either himself or Osama Bin Laden.
But the BBC Afghanistan correspondent says the opposition is hoping for widespread popular discontent that the Taleban might not be able to cope with.
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