The insurgent group said the ongoing assault began with a suicide attack targeting local and foreign intelligence buildings at a roundabout in the city.
AFP reporters heard a string of loud blasts while police confirmed one explosion and a gunfight close to the heavily-guarded embassy compound.
An AFP reporter close to the scene saw a string of mortar rounds fired and small arms fire, and heard explosions that seemed to hit multiple targets.
He said that police and Afghan military were attempting to approach the roundabout but were retreating under fire.
"Today at one o'clock at Kabul's Abdul Haq roundabout a massive suicide attack on local and foreign intelligence facilities is ongoing," said a spokesman for the insurgent group, Zabiullah Mujahid, in a text message to AFP.
A Western military source confirmed NATO's International Security Assistance Force headquarters was one of the targets under attack.
"ISAF HQ is under attack at the moment," the source said.
An ISAF spokesman would not confirm the headquarters was a target but said: "There is an ongoing attack in the centre of Kabul."
One eyewitness reported that attackers had taken up position in a tall building under construction and were exchanging fire with security forces. An Afghan National Army installation is nearby, as is a Marriott hotel building site, the witness said.