The assault came just a day before President Ashraf Ghani is due to meet world powers at a major donors conference in Brussels.
The attack began in the early hours at the southern and eastern approaches to the provincial capital where the militants were engaged in battles with government forces, an AFP correspondent said.
Two Afghan army helicopters were flying over the city, which was deserted, with streets empty and shops closed.
Government control of the city has been shaky ever since.
Ghani will meet with world leaders in Brussels on Tuesday and Wednesday in a bid to secure financial aid from the international community up to 2020 to rebuild his war-ravaged country.
The meeting will try to drum up support from an international community suffering from aid fatigue as it grapples with conflicts in Syria and Iraq plus the worst migration crisis since World War II.
The conference comes as Afghanistan struggles to negotiate peace with a resurgent Taliban and other militant groups who continue to wage a bloody insurgency nearly 15 years after the US invasion.
In September, Kabul signed a peace deal with notorious warlord Gulbuddin Hekmatyar, who heads the largely dormant Hezb-i-Islami militant group, a move that will have little impact on security but is a symbolic victory in efforts to bring the Taliban to the negotiating table.
"Nobody can afford for Afghanistan to destabilise again," a senior EU official added.
After seizing Kunduz on September 28, 2015, the Taliban held the city for two days and eventually announced they were withdrawing from the outskirts on October 15.
The United Nations said that battle left 289 people dead and hundreds more wounded.
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