The fighting is in an area around the presidential palace, Serena Hotel, the Central Bank and Ministry of Justice.
The Taliban says 20 fighters, including suicide bombers, are involved. Nato forces say at least two insurgents have been killed.
This is the latest in a series of increasingly brazen attacks on Kabul.
A statement on a Taliban website said the latest raid had specifically targeted government buildings and the hotel.
There are reports of one recent massive explosion in the diplomatic quarter
'Chaotic scene'
Nato's Isaf force said it was "working closely with our Afghan partners to aggressively contain the situation during which several small explosions were reported near the Feroshgah-e-Afghan shopping centre and the Serena Hotel".
Streets in the centre of the city have been closed and hotels and government buildings have been locked down.
There are reports of a number of wounded people being ferried away in ambulances.
One official trapped in a government building told the Reuters news agency: "It is a chaotic scene, we do not know what to do and where to go."
Taliban militants have launched a number of recent attacks on Kabul.
In October, five UN staff were killed in a raid on a UN guesthouse. The Serena Hotel was also targeted in the attack.
The BBC's Mark Dummett in Kabul says there will be huge concern that the militants have again broken through to the most protected part of the city, although security forces say they do prevent many other attacks.
Monday's attack comes amid continuing political uncertainty in Afghanistan.
Some new members of President Hamid Karzai's cabinet were being sworn in at the time of the raid, but there are still a large number of posts vacant.
Parliament has twice rejected many of Mr Karzai's nominations for a new cabinet, forcing the president to direct deputy ministers or other caretaker figures to run their ministries.
The uncertainty comes ahead of the key London Conference on Afghanistan later this month.
Mr Karzai was re-elected last August in a vote marred by fraud.
Since then, the US and Nato have launched an overhaul of Afghan policy, with a large number of additional troops pledged.
