Early reports of the attacks, which began on Monday close to the presidential palace, said that at least four people had been killed and that several people had been injured.
Farhad Paiker, an Afghan journalist, told Al Jazeera that the strikes had spread to other parts of the city.
"A suicide bomber in a car came towards the foreign ministry. Security forces tried to stop it and it hit a shopping centre," he said.
"It is really chaotic in the area."
Later a car bomb exploded near another shopping centre, close to the education ministry.
A security source was quoted by the Reuters news agency as saying that a suicide bomber had killed "several police and intelligence officials".
The Taliban have claimed responsibility for the attacks, saying that 20 of their fighters were involved in the clashes.
Afghan security forces have locked down the centre of the city. Nato's international forces were also reportedly helping to secure the area.
"We can confirm that four suicide bombers have been killed," Zemarai Bashery, an interior ministry spokesman, told Al Jazeera.
He said that two of the attackers had taken refuge in a building and had been killed by security forces.
Qais Azimy, Al Jazeera's Kabul producer, reporting about 200m from the secene of some of the fighting, said: "There are hundreds of Afghan army and police and intelligence officials present. Civilians have completely left the area.
"It is a big question mark how they [the fighters] got so close to the presidential palace."
David Chater, Al Jazeera's correspondent who was at the Serena Hotel surrounded by the fighting, said: "We've heard four large explosions very close to the hotel. I'm told the fighting is close to the presidential palace.
"There is fierce gunfire very close to me ... a heavy gunfight going on at the moment.
"It is extraordinary that security has been breached to this extent," he said, adding that it showed the Taliban could act at will.
The Serena Hotel, which is frequented by foreign journalists, was reported to be on fire shortly after the attacks began.
Chater said there had been recent warnings that opposition fighters had hijacked six armoured cars and were planning an attack on the capital.
The attack appears to be the most co-orindinated offensive on the capital since the US-led invasion in 2001 that toppled the Taliban from power.
But Afghan politicians insisted the Taliban attack held little significance.
"It does not mean the Taliban is strong. It means they can not target military targets only the Afghan people. It means that they are weak."
Even so, the attack comes at a sensitive time in Afghan politics with Hamid Karzai, the president, yet to finalise his cabinet after disputed elections.
Chater said that the strikes will bring into question the credibility of Karzai's authority and the military strategy in Afghanistan of Barack Obama, the US president.
Obama committed 30,000 extra troops to the country at the end of last year, to be focused on training local security forces, after much deliberation.
The attack also comes ahead of the London Conference on Afghanistan hosted by the UK, UN and Afghanistan on January 28 on winning the conflict in the country.