De Luca, capped 38 times, suffered the injury in training with his club side Edinburgh in December and will be out for around two months.
"The damage required time to settle for full assessment," Edinburgh said in a statement.
"It was deemed that the operation would be the best course of action to support the quickest return to playing,"
Scotland open their Six Nations campaign against England at Twickenham on February 2. They host Italy at Murrayfield a week later, Ireland on February 24 and Wales on March 9. Their final game is away to France on March 16.