One blast hit the area near a shrine and an hour later another occured near the residence of the governor of the Nangarhar Province.
The attack occured at 8:30 a.m. local time (3:30 GMT).
According to reports, one of the attackers detonated his suicide vest in a line of people waiting to receive their salaries at a private bank, 60 meters away from the scene of the first explosion.
The Ministry of Internal Affairs condemned the attacks and said security forces located another two explosive devices near the scene of the first blast and detonated it.
One more explosion of a magnetic bomb on a civilian car took place in the province's Behsood district later, killing one person and injuring another.
A spokesman for the Islamic State in the region said the group took responsiblity for the blast at the shrine.
In their turn, Taliban rejected any responsibility and condemned the attacks.
However, earlier the group warned of assaults due to the presence of foreign military in Afghanistan.
The wounded were hospitalized. Many of them are in critical condition.
The Afghan parliament members passed their condolences to the victims' families.