Taliban fighters have seized a military hilltop site in Kunduz, tightening their grip on the northern Afghan city.

The capture of the Bala Hisar fortress came despite efforts by government reinforcements, backed by Nato airstrikes and special forces, to retake the city.

It leaves the airport as the army's last stronghold.

The Taliban overran Kunduz on Monday, their biggest military gain since they lost power in 2001.

The Taliban had blockaded the Bala Hisar fortress for two days. Hundreds of Afghan security personnel were stationed there, and their fate was not clear.

Late on Tuesday the Taliban tried to capture the airport as well, but two US air strikes halted their advance.

Clashes went on through the night around Kunduz, and the provincial hospital was reportedly struggling to cope with the number of casualties.

Kunduz is one of Afghanistan's largest cities, and is strategically important as a transport hub for the north of the country.

Militant violence has increased across Afghanistan since Nato ended its combat mission in Afghanistan in December, leaving a 13,000-strong residual force used for training and counter-terrorism operations.

Fighting has been going on in a number of other areas:

- In neighbouring Baghlan province a former Nato base was reported to be under attack, while government forces had been blocked by Taliban-laid mines

- Reports speak of fighting in several districts in Takhar province, to the east of Kunduz

- The Taliban say they have taken a district in western Farah province.

The BBC's Dawood Azami says the Taliban are now trying to open multiple fronts to divert the attention of the Afghan military from Kunduz and stretch them thin.

On Tuesday the United States acknowledged the seizure of Kunduz as a setback, but said it remained confident that Afghan security forces could retake the city.

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."

There was no independent confirmation, and the Taliban denied their local leader had been killed.

The Afghan defence ministry claimed on Tuesday that 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 featured militants in the town showing off seized tanks, armoured vehicles, police cars and Red Cross vans.

Residents, nervous of both the Taliban and the possibility of street-fighting in the battle for the city, are largely staying indoors.

An Afghan MP, Dr Fatima Aziz, told the BBC she fled the city with her children, coming under fire from the Taliban as she escaped.

"I'm very worried for my people," she said.

The Taliban's new leader, Mullah Akhtar Mansour, said the government should admit defeat.

In Kabul, members of parliament called for President Ghani to resign. The head of Afghan intelligence was called to parliament to explain what went wrong in Kunduz.

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 before moving into the centre.

Kunduz province has seen a number of attacks since April, with the Taliban joining forces with other insurgents.