Lidl is working with construction firm Dickie & Moore to renovate the store which will create up to 40 new jobs when opened.
The fit out is expected to take three months, with a view to opening the store in November this year.
Gordon Rafferty, Lidl UK’s regional head of property, said: “There’s been much anticipation for this new Lidl store and it’s great that we are now in a position to start construction to convert the building.
“We are extremely grateful for all the support that we have received so far, and would like to thank everyone for their patience and understanding whilst we enter this next phase of the development.
Gordon added: “We look forward to bringing a discount option to the town of Giffnock before the Christmas rush.”