At least 11 people are reported injured so far.
A suspected suicide car bomb blast began the assault, with the gunmen now said to be using machine guns and RPGs from higher floors in the building.
Images showed residents fleeing from the area. No group has yet said it is behind the attack.
Jalalabad, near the border with Pakistan, is often targeted by Taliban militants but it is also a stronghold for the Islamic State group, whose fighters have been active there since 2015.
The latest attack comes days after Taliban gunmen stormed a luxury hotel in Kabul killing at least 22 people, mostly foreigners.