The company announced plans that would see 37 jobs go, leading to cost savings of £1.5m a year.
Key to the proposals is the proposed closure of the Barrow print plant, where 28 jobs are under consultation.
Other threatened positions involve telesales and consultancy roles.
According to Unite, the work would be moved to the company's Carlisle plant.
The move comes as a surprise following a £5m investment in a new press for the Barrow plant just five years ago. Furness also installed the world's first Ferag Rollsert machine.
CN Group was unavailable for comment, however chief executive Robin Burgess told The News and Star, which is owned by CN Group: "The original and proposed changes have been done in such a way to ensure all the group’s titles are still being published and there is a minimal effect on content.
"It is very sad to be making this announcement today, but if the proposals go forward it will ensure we are well placed to withstand the recession and take advantage of the recovery when it comes."
28 jobs under threat at newspaper print site

The Barrow print site of North West newspaper printer Furness Newspapers faces closure as part of a planned restructure by parent company CN Group.