In a sign of violence intensifying across the country, fighting also raged in the southern province of Helmand, where insurgents overran the district of Nawa, just south of the provincial capital, and killed the district’s police chief in an overnight attack.
Since the NATO coalition drew down to an advisory contingent of about 10,000 troops, Afghan forces have struggled to contain the Taliban, losing territory to the insurgency over the past year and suffering record casualties.
The coordinated attack on Kunduz, coming from four directions, began before dawn on Monday, according to Mahfozullah Akbari, a spokesman for the regional police zone. Fighting continued in at least three parts of the city as shops remained closed and residents tried to flee. Helicopter gunships were also seen targeting Taliban areas, some less than a mile from the governor’s compound.
The Afghan Interior Ministry and the American-led NATO mission in Afghanistan played down fears that the city was on the verge of falling again.
“We are aware of reports of ongoing sporadic fighting in Kunduz and are coordinating closely with our Afghan partners to assist,” said Brig. Gen. Charles H. Cleveland, a spokesman for the international coalition. He said they were not seeing evidence “to support the reports that Kunduz is under significant attack.”
Civilians were again caught in the line of fire, as residents reported that both the Taliban and Afghan forces used their homes to fire on the other side.
Sardar Murady, a resident who lives close to the highway leading to the district of Chardara, said the Taliban were using some civilian homes for fighting.
“They told us not to lock the gates to our houses,” he said.
Kabir Shabaan, a resident of the Seh Darak area near the city’s southern gate, said Afghan forces asked them to leave their homes, as it was the front line.
“They told us we need to turn this into a trench,” Mr. Shabaan said.
Almost exactly a year ago, insurgents briefly overran Kunduz, making it the first urban center to fall to the Taliban since the collapse of their regime in 2001. During the operation to retake the city, American planes mistakenly bombed a hospital run by Doctors Without Borders, leaving at least 42 people dead.
In Helmand, officials said the capital, Lashkar Gah, also remained largely surrounded, and the fall of Nawa had only added pressure from an additional direction.
Abdul Karim Attal, the leader of the provincial council, said fighting was continuing in six areas around the city.
“Lashkar Gah is surrounded from all four sides, and Taliban are staging attacks from all four sides,” he said. “I urge the central government to halt the rapid progress of the Taliban — if the reinforcements do not arrive in days to Helmand, we will be experiencing the fate of Kunduz soon.”
Nawa, which fell to the Taliban on Sunday night, was under constant fire for four days, said its governor, Aqa Muhammad Takra. He said that he lacked enough forces and ammunition and that the government had not properly planned the defense of the district after it barely survived an onslaught by the Taliban in August.
“The government helped break the previous siege but did not hold the area permanently by installing check posts on the main road and putting more forces in the district,” Mr. Takra said. “It became the focus of the Taliban again, and we haven’t seen any reinforcement to protect the district from falling.”
In a sign of the disarray in the Afghan defense ranks, repelling the onslaught in August required a delegation of senior generals and officials sent from Kabul to shuttle back and forth to monitor developments. Once the Taliban were pushed back — after leaving government buildings badly damaged and blowing away security towers — the generals and officials were taken by helicopter to help the neighboring district of Garmsir.
Nawa’s police chief, Col. Ahmad Shah Salem, who was central to the district’s defense during the assault in August, was killed in the attack on Sunday night, said his brother Abdul Wadud, a member of Parliament.
Those who remained of the roughly 150 police officers and 50 soldiers in Nawa were stranded near a river, Mr. Wadud said, and the government was struggling to rescue them.
The death of Colonel Salem was the latest blow to an Afghan security force that has had record casualties this year. While the government does not release death tolls, figures for a couple of months this year suggest an increase in fatalities in the security forces over last year’s total, which was estimated at 6,000.
In July, 900 Afghan soldiers and police officers were killed, according to the NATO commander in Afghanistan. In August, perhaps the deadliest month so far, President Ashraf Ghani told a group of civil society activists that 1,000 members of the Afghan security forces had been killed, according to Khan Zaman Amarkhail, who attended the meeting.
Nader Nadery, a senior adviser to the president, said that figure included some civilians as well.
“In August, the Taliban’s mobilization of all resources was unprecedented — and it was beyond their capacity,” Mr. Nadery said. “This is no longer an undeclared war — this is a declared war, and the Taliban had full-fledged support not seen before,” he added, referring to the long-held belief on the part of Afghan and Western officials that the military in neighboring Pakistan is aiding the Afghan insurgents.
A previous version of this article included an incorrect dateline. Najim Rahim reported from Kabul, Afghanistan, not Kunduz.