Mahfoz Akhbari, spokesman for the chief of Kunduz police, said battles continued in and around the city where Taliban fighters were using civilian homes to launch attacks.
Forty Taliban fighters have been killed, he added.
Fatima Aziz, a member of Afghan parliament from Kunduz, described the security situation as "very bad" on Tuesday morning.
"They [Taliban] are coming into people's houses all of the night -- they want a place to stay under the house. The Taliban is taking food from families," Aziz told CNN.
Faisal Noori, a local television reporter based in Kunduz, told CNN Tuesday evening that gunfire could still be heard "everywhere in the city."
"All the shops are closed down. People do not have enough food to eat and water to drink. Even the electricity is cut," said Noori.
Sediq Sediqi, spokesman for the Ministry of Interior, said Tuesday that more than 100 special forces had been deployed to clear Kunduz of Taliban militants in an operation that was still ongoing.
The attack on Kunduz comes as President Ashraf Ghani is in Brussels for an important aid meeting where he will attempt to secure funding from the international community and defend his track record on reform.
On Monday it was unclear which parts of the city the Taliban had seized.
Shukria Paiman Ahmadi, a member of parliament who represents the area told CNN, "The Taliban have overrun important parts of Kunduz city and taken control of the city center." She was in the capital, Kabul, but had spoken to people in the city, she said.
A tweet from the US-led NATO mission Monday said there was "increased Taliban activity" in Kunduz. On Tuesday, it said the government was in control.
Earlier Monday, both Afghan and US officials had downplayed the threat.
"This time, the Taliban won't be able to capture any part of Kunduz," said Mahfoz Akbari, spokesman for the police in Kunduz, told CNN by phone.
US Brigadier General Charles Cleveland characterized the current clashes in Kunduz as "ongoing sporadic fighting."
"We are not observing evidence via our internal means to support the reports that Kunduz is under significant attack," he wrote in an email to CNN.
Memories of 2015 attack
A Taliban spokesman and Afghan government officials gave similar descriptions of how militants launched a pre-dawn attack on the city from four different directions.
"Our Mujahideen are advancing rapidly inside Kunduz," Taliban spokesman Zabiullah Mujahid claimed in a statement to CNN.
An aid worker said that memories of Taliban attack in 2015 had spurred huge numbers of people to flee.
"A lot of people who fled Kunduz last year at this time when the Taliban had taken over the city returned to Kunduz. So the number of people trying to flee again today was huge," Ehsanullah Sadiqi told CNN Monday, speaking by phone from Kunduz.
Longest foreign conflict in US history
The US is approaching the 15th anniversary of its war against the Taliban in Afghanistan. It is the longest foreign conflict in US history.
The US military is reducing its footprint
on the ground to roughly 8,500 troops from peaks of more than 100,000 servicemen and women just a few years ago.
But US forces have carried out more than 700 airstrikes in the first eight months of 2016 in large part to support Afghan troops on the ground.
Despite this air support
, the Western-backed Afghan military is suffering some of its worst casualties yet.
"We believe that there has been a 20% increase in ANDSF [Afghan National Defense and Security Forces] casualties, so far this year compared to this time in 2015," wrote Colonel Michael Lawhorn, Director of Public Affairs for US Forces Afghanistan, in an email to CNN.