Heavy fighting between Afghan security forces and Taliban insurgents was reported around the northern city of Kunduz on April 27.
Officials said hundreds of Taliban militants had attacked police and army checkposts in the province of Kunduz.
Now they threaten to overrun parts of the provincial capital, after fighting that killed eight Afghan security personnel and at least two dozen Taliban, a spokesman for the provincial governor said.
"The threat level is very high, but with new reinforcements, our security forces have gained morale," said Abdul Waseh Basel, the spokesman.
Over the past three days, the northern Kunduz Province has seen large-scale Taliban attacks.
The province's governor, Omer Safi, said reinforcements have arrived and that plans are under way for an operation in three different parts of Kunduz.
"There are casualties as a result of three days of battles on both sides, but I can't provide precise figures as the fighting is still going on," Safi said.
The fighting led President Ashraf Ghani to delay his departure on a state visit to India by several hours.
He had delayed his departure for New Delhi to meet the top NATO commander in Afghanistan, U.S. General John Campbell.
Fighting continued into the evening, but Afghan forces pushed the Taliban back with help from reinforcements from Kabul and other areas, said local army commander Qadam Shah Shaheen.
Militants this month launched major attacks in another northern province, Badakhshan, and on April 27 fired on a government delegation meeting soldiers there.
Their rockets and gunfire narrowly missed the group, led by Ahmad Zia Massud, head of Ghani's governance commission, a close aide said. The delegation withdrew by helicopter.
The Taliban has stepped up its attacks against Afghan security forces across the country and last week launched the start of its annual spring offensive.