Defender Elphick, 28, first sustained the injury during the 4-3 win against West Ham in August.
The former Brighton player last appeared for the Cherries in their 3-1 defeat at Norwich last month.
"Tommy has had surgery on his ankle and could face 10 weeks out. It could be shorter," said manager Eddie Howe.
Elphick was ever-present in the league last season as he captained the Cherries to the Championship title.
Howe also confirmed striker Callum Wilson has had an operation after he ruptured the anterior cruciate ligament in his right knee against Stoke.
Wilson, 23, faces up to six months out.
Winger Ryan Fraser, currently on a season-long loan at Ipswich, will receive treatment at Bournemouth as he also recovers from medial ligament damage.
The 21-year-old sustained the injury while training with Scotland Under-21s and is set to miss the next two months.
Bournemouth captain Tommy Elphick has been ruled out until the new year following an operation on an ankle injury, the club has confirmed.