The Afghan Interior Ministry said one policeman had been killed and another four wounded in the ongoing fighting. It said in a statement that the situation is being monitored in case reinforcements are needed.
Kunduz is the capital of the strategically important Kunduz province, a breadbasket region that borders Tajikistan to the north and sits on a major crossroad in the country.
The city was overrun by Taliban in September 2015, sending shockwaves through the country as it was the first time the militant group had taken a major urban center since launching the insurgency 15 years ago.
Mahmood Danish, spokesman for the provincial governor in Kunduz, said the insurgents staged today's attack from different directions but that security forces managed to keep them at bay.
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The Taliban were using residential areas in the assault and "our security forces are being very careful to avoid civilian casualties while shooting back at the enemy," he said. The Afghan air force was also supporting the ground forces in the fight, he added.
Taliban spokesman Zabihullah Mujahid claimed the insurgents had captured several checkpoints in the city.
The US military spokesman in Afghanistan, Brigadier General Charles Cleveland said the situation in Kunduz was being monitored but added: "We are not observing evidence via our internal means to support the reports that Kunduz is under significant attack."
Later, Mohammad Yusouf Ayubi, head of the Kunduz provincial council, said heavy battles were taking place and that government offices, schools and shops have all been closed.
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