Donnie has not played for Glasgow since the match against Edinburgh at Murrayfield last December.
The club now hopes the operation on his left knee - carried out by the same surgeon who repaired team-mate Jon Petrie's dislocated shoulder - will allow him to return to full training with the squad as soon as possible.
Warriors head coach Hugh Campbell said to Sporting Life: "Donnie had his knee tidied up by the same surgeon who looked after Jon Petrie's shoulder injury.
"It all went well and we hope he'll be back into full squad training very soon."