However, Afghanistan's intelligence agency blamed the Taliban's Haqqani network and Pakistani group Lashkar-e-Taiba for both attacks.
At least 6 people were injured, informed the press-Secretary of the Ministry of health Wahidullah Mirah, adding that the total number of victims is likely to grow.
After an easing of violence in Kabul in February and March, Taliban and IS militants have stepped up attacks in the city in recent weeks.
The blast and gun battle in PD13 was followed by one more explosion in Shahr-e-Naw area in Kabul's PD10, according to the sources. Barmak mentioned "two or three" extra attackers have been holed up in a close-by constructing, buying and selling hearth with safety forces.
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. According to the eye witnesses, the first suicide bomber was on a motorcycle, who detonated himself among the crowd.
Taliban spokesman Zabihullah Mujahid announced that the district centre and police headquarters in Tala Barfaak had been overrun by the group's fighters on Tuesday.
Earlier this month, an attack in Kabul that included a secondary explosion in a crowd that gathered after a first explosion, killed more than 25 people, including nine journalists.
Estanakzai told AFP that surrounding buildings had been evacuated but it was not clear if people were trapped inside with the attackers.
Smoke rises from the site of the second deadly suicide attack on police stations in Kabul, Afghanistan.
A travel agency that handles Indian visa applications is located on the same street as the Shar-e-Naw police station. It was unclear who was behind the assaults, however the Taliban and an Islamic State affiliate are each lively within the province. "Had the police not stopped the attackers they would have caused a disaster", Barmak said. "Such blatant terrorism targeting innocent civilians and disrupting way of life has no justification whatsoever", it said, reiterating that Pakistan stands ready to extend all possible counterterrorism cooperation to Afghanistan. Those attacks were claimed by IS.
One of the bombings was claimed by ISIS and the other by the Taliban.
One blast took place near a police station.