It is thought that the data included their roles and salaries, as well as personal information like bank account details and National Insurance numbers.
The company employs around 13,000 people across 35 locations all over the UK and Ireland.
Specialist data breach law firm Hayes Connor is now working with over 35 individuals affected by this incident and is receiving new enquiries on an almost daily basis.
Now, the lawyers are calling for Greencore to work quickly to provide further information on what has happened.
Christine Sabino, the legal specialist at Hayes Connor who is representing those affected, said: “The information we have received is hugely concerning and further answers are clearly needed.
“Greencore claims there is no evidence that data has been misused, but there is no way to tell for certain that this is the case. No guarantees can be provided about the future either.
“This company employs thousands of people across a range of sites, but no real indication has been provided on how many have been affected. While we have heard first-hand from a number of people worried by these developments, there will likely be many more who are also concerned about what has happened.
“Employers have a duty to ensure that incredibly sensitive information is kept safe and secure, so this type of incident warrants a significant investigation. We have started to make our own enquiries into the case and are determined to ensure that our clients get the answers and justice they deserve.”
Greencore has been asked to comment.