The action is being taken over alleged damage caused by employees when they occupied MMP's Bootle factory last month.
Meanwhile, union members are continuing their tour of sister MMP plants, with three representatives in Auxerre, France today.
Workers at the Bootle site initially went on strike over redundancy terms and selection criteria on 10 February.
When a second strike was attempted the following weekend, management instigated an employee lock-out, although this led staff to occupy the building.
Unite national officer Ian Tonks said: "The company has started going through disciplinary procedures over incidents that are alleged to have taken place inside the factory.
"We are still hoping to sit down and talk to them. They have told me they are not available to talk at this time, but they will come back to me."
Management at MMP was not available for comment while the Bootle plant has not produced work for almost four weeks.
Employees from MMP Bootle are meeting with colleagues in France today to explain their side of the story regarding the dispute.
Tonks added: "When we went to Germany the first time, we found the workers view had been skewed by management; they had a very different view to us. We are meeting with these colleagues to put our side of the story across.
"It is workers talking to workers: they are just a group of employees, exactly the same as the guys in Germany or in France. The only difference is the language that they speak. I think they are quite pleased to be put right on what has occurred."