There were no immediate reports of casualties.
Two police officials say the car bomb exploded outside the embassy, followed by an attempt by gunmen to enter the building, which is located in the center of the Afghan capital.
The firefight is continuing as witnesses in the area reported hearing gunshots.
The two Interior Ministry officials spoke on condition of anonymity because they are not authorized to speak to the media.
No one has taken responsibility for the attack, though both the Taliban and an Islamic State affiliate have previously carried out such attacks in Kabul.
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