There were two explosions near the city's directorate of finance, Nangarhar provincial governor's spokesman Attaullah Khogyani told AFP, adding that "a number of attackers" had entered the building.
"Security forces are in the area chasing and fighting them," he said.
Four bodies and about 20 wounded people had been brought to Jalalabad hospitals so far, said Dr Najibullah Kamawal, director of the city's health department.
There was no immediate claim of responsibility for the attack.
Jalalabad is the capital of restive Nangarhar, which borders Pakistan. Some areas of the province are a stronghold of the Islamic State group but Taliban fighters are also active there.
It was the latest deadly violence to strike Afghanistan as militant groups step up attacks and US-backed Afghan forces intensify air strikes and ground offensives.
The Taliban have rejected growing calls to accept President Ashraf Ghani's offer of talks to end the 16-year conflict.
