The incident is said to have happened in the Shar-e Naw neighborhood in the north-west of the city and a gun battle is under way. A bomber blew himself up at the gate of the Embassy while three others stormed the compound.
The attack began with an explosion outside the embassy, authorities said. The Interior Ministry did not give details regarding casualties but said the Embassy staff were moved to a safe location.
The Taliban, fighting to reestablish strict Islamic law 16 years after being expelled by a US -led campaign in 2001, have opposed Islamic State.
A police spokesman said security forces were at the scene and a gunbattle was under way.
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Islamic State has carried out a series of high-profile attacks in Kabul, mainly targeting members of the mainly Shi'ite Hazara community, and fuelling concerns of a possible spillover into Afghanistan from fighting in Syria and Iraq.
Images on social media show a great plume of thick black smoke rising over the area.
The attack is the latest to rock Kabul, and comes as the resurgent Taliban ramp up their offensive across the country during the warmer weather fighting season.
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They claimed they would be "ready for dialog" with Qatar if it met a series of demands they have presented to Doha. Armed confrontation between the two countries had been narrowly avoided in 1986 due to timely Saudi intervention.