"Allan has damaged the pectoralis major muscle adjacent to his right shoulder. It will require repair and the operation will be carried out by surgeon Richard Nutton at the Spire Murrayfield Hospital," said Scotland and Lions doctor James Robson.
Glasgow Warriors skipper Ali Kellock also picked up an injury in the win over the Italians, although Robson believes he should be able to resume training at the start of next week.
Kellock strained ligament sin his left ankle and has had his left foot put into an immobilising boot.
There was also good news for Kellock's Glasgow team mate Max Evans as X-Rays showed he has nothing more than severe bruising to his left cheek.