Earlier this week, the group revealed that nearly half of the 120 print jobs set to be cut could be retained within the company.
The union has been fighting the cuts since CUP originally announced the start of a 90-day consultation process with staff at its UK printing arm, Cambridge Printing Services, on 22 January.
"We are, of course, pleased with the progress but in our eyes this is not the final outcome. We will continue in talks with the university and our members with the next meeting taking place on 2 April," she said.
Options being explored by CUP include possible redeployment for some of those affected. In addition, it is considering maintaining the print operations from other parts of its publishing arm until other solutions can be explored.
Peter Davison, corporate affairs director at CUP, said it now aims to print a higher proportion of four-colour European journals, which would have previously gone elsewhere.
He said: "We have had some very constructive discussions so far and we look forward to working with them until the end of the consultation to ensure that we get the best possible deal both for our staff and for the business."
Unite reveals 'progress' made at CUP job talks
Progress is being made over proposed job cuts at Cambridge Universtity Press' (CUP) print arm, Unite national officer Ann Field has said, but added that there is still much to be done.
