Police are investigating if the attack is linked to the Evening Post’s extensive coverage of the Bristol disturbances.
The Post has published several photos of suspects believed to be involved in Monday’s violence, in which a Cabot Circus jeweller was ransacked.
At the front of the four-storey block, windows were smashed and paint was thrown by vandals.
Staff at the newspaper are at work as normal, but a sign at the front of the building asks them to avoid using that entrance.
Paul Breeden, chair of Bristol NUJ, said: “The NUJ condemns any attack on journalists or the premises where they work. There is no justification for violence.
“If it emerges that the damage to the Post and Press building was connected to coverage of the riots, I hope that the law will take into account the importance of having a free press that is able to report on its community without persecution.
“Journalists from the Evening Post, from the BBC, from the website Bristol 247 and elsewhere have been reporting on the recent disturbances sometimes in very difficult circumstances.
“Without them people would not have a picture of the events that have shocked our city and the nation. Journalists deserve public support – there is never any justification for attacking them or the premises from which they work.”