But despite repeated government pledges to tighten security, hundreds of people have been killed and wounded in attacks in the city since the beginning of the year and authorities have appeared powerless to stop the bloodshed. Several other attackers were outside the building shooting at police officers.
The eight suicide bombers involved in the two attacks struck two police stations in Kabul.
Barmak said "two or three" more attackers were holed up in a nearby building, trading fire with security forces.
One journalist shared a gruesome image on a WhatsApp reporters group showing what appeared to be a suicide bomber who had detonated his explosives but only killed himself.
Asam, the head of the Kabul ambulance service, who only has one name, said six people have been transferred to hospitals.
The Taliban issued a statement saying they controlled the district and claiming to have killed 10 security forces.
Hashmat Stanikzai, spokesman for the Afghan National Police in Kabul, told Anadolu Agency the explosions took place at Police Station 13 in western Dasht-e-Barchi neighborhood and close to Police Station 10 in Shahr-e-Nau area of the city.
The Islamic State outfit claimed responsibility for the attack in Police District 13 while the Taliban reportedly claimed responsibility for the other attack, Xinhua said. Twin suicide bombings claimed by IS last week killed at least 25 people, including nine journalists who had rushed to the scene of the first attack.
An Afghan lawmaker says Taliban fighters have captured a second remote district compound in two days.
The Taliban and Daesh frequently target Afghanistan's Western-backed government and security forces.
Mohammad Hashim, a member of parliament from Faryab province, said the Taliban captured the district headquarters in Bilchirgh early on Wednesday after more than 40 security forces retreated under heavy fire.
The Taliban on Tuesday captured another district compound in northern Baghlan province.