Paragon acquired the business after purchasing the assets of the label division of William Sessions, which traded as Sessions of York, out of administration on 21 April 2010.
Paragon intends to relocate the business to its digital print facility, which is situated in Boston, Lincolnshire.
Mark Lapping, Paragon group managing director, told trade magazine Packaging News it was a "sad day" as the company had wanted the business to remain in York.
He said: "It's a consultation process and we can't pre-empt what will happen. But if the move goes ahead, we will be assisting those staff who want to move and we will look after those who don't."
Paragon has a turnover of £135m and employs around 950 staff. Sessions' label business has a turnover in the region of £4m.
"We will consider all proposals put forward by employees during the consultation period to ensure we make the right decision. I would like to assure customers that production will not be affected during this process," Lapping added.