Witnesses to Radio Free Europe (RFE) said the bomb targeted the vehicle carrying the victim.
Local Taliban officials did not disclose details of the incident.
A doctor at Asadabad Central Hospital, who requested anonymity, told RFE that 12 people injured in the blast had been hospitalized. Doctors confirmed that Abdullah Irfan, the Taliban-appointed police chief from Shigal district, had died.
“Seven civilians, including a schoolboy, were also among the injured,” the doctor added.
No group has claimed responsibility for the attack but the Islamic State group Khorasan (ISIS-K), which is active in eastern Afghanistan, has claimed similar attacks against the Taliban in the past.
The Taliban itself has been fighting ISIS-K since its emergence in Afghanistan in 2014.
ISIS-K has claimed responsibility for several recent attacks, including last week’s suicide bombing among worshipers at a Shi’ite mosque in Kunduz province that killed around 100 people some time ago.