Last month, the group revealed that nearly half of the 120 print jobs set to be cut could be retained within the company.
The 90-day consultation with staff is set to end at the end of this month with Unite national officer Ann Field placing securing employment a priority from the remaining talks.
"The next step is to press for more jobs to be retained and we won't rest until we have secured more employment. If not, we want decent redundancy packages for those affected," she said.
In order to save some of the positions at CUP, options being explored include possible redeployment for some staff. In addition, the press 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 has previously said the university now aims to print a larger number of its four-colour European journals, which would have previously gone elsewhere.
"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," he added.
Unite seeks to save more jobs at CUP
Unite has vowed to "press for more jobs" to be retained at Cambridge University Press as both parties plan to meet again later this week to continue talks concerning the future of the print arm.