Taliban spokesmen were not immediately reachable for comment.
There was no initial claim for the blasts.
Jalalabad is the capital of Nangarhar province, which is a stronghold for the Islamic State (IS) militant group that has been active since the fall of Kabul to the Taliban.
A series of bombings at Kabul airport on 26 August that were claimed by IS killed more than 180 people. Those attacks were among deadliest during the country's two decades under US occupation.