Two US air strikes on Tuesday halted an attempt by insurgents to seize the airport, the army's stronghold.
Afghanistan's spy agency says the strikes killed the Taliban leader in the province and his deputy, but the Taliban has denied this.
The capture of Kunduz represents the militants' biggest victory since 2001.
On Tuesday the United States acknowledged the seizure of Kunduz as a setback, but says it remains confident that Afghan security forces can re-take the northern city.
Overnight there were clashes around the city, but it remains unclear just how much of it remains under government or Taliban control.
Sayed Sarwar Hussaini, a spokesman for Kunduz's police chief, told Reuters: "Hundreds of Taliban are killed and their dead bodies are on (the) streets."
The Afghan defence ministry claimed the town's police headquarters and prison had been recaptured, after militants released hundreds of prisoners when they took the city on Monday.
But Taliban-released video showed militants in the town showing off seized tanks, armoured vehicles, police cars and Red Cross vans.
An eyewitness told the BBC that Taliban reinforcements had also arrived, with the situation too dangerous for locals to leave.
Residents, nervous of both the Taliban and the possibility of street-fighting in the battle for the city, are largely staying indoors.
The Taliban's new leader, Mullah Akhtar Mansour, said the government should admit defeat.
President Ashraf Ghani, who completed his first year in office on Tuesday, said in a televised address that "progress" was being made recapturing Kunduz, but security forces had been hampered by the Taliban using civilians as human shields.
The assault on Monday was swift and took Afghan forces by surprise.
As darkness fell, heavily armed fighters crossed fields to attack the city from multiple directions. They quickly overwhelmed several of the police checkpoints defending the perimeter of the town.
They then captured key buildings, freed about 500 prisoners from the city's jail and forced officials and troops to retreat to the airport.
Kunduz province has seen a number of attacks since April, with the Taliban joining forces with other insurgents.
Nato ended its combat mission in Afghanistan in December, withdrawing most of its troops, apart from a 13,000-strong residual force used for training and counter-terrorism operations.
Militant violence has increased across Afghanistan since the departure of most Western forces.
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