Jones, 35, sustained the injury during the Lions' warm-up game against Japan at Murrayfield on 26 June, but he will fly into Cape Town on Thursday.
The veteran lock came through a training session with the Wales squad at their Vale of Glamorgan base this week before successfully undergoing a medical assessment.
"We are delighted to welcome Alun Wyn back," said head coach Warren Gatland. "It'll come as no surprise to anyone who knows Alun Wyn that since injuring his shoulder against Japan, he's done everything he can to get himself back.
"It's remarkable really when you consider it's just 18 days since he left us in Edinburgh.
"He's been training with the Wales squad at the Vale since last week and yesterday he had a proper hit out. Following assessment from the medical staff this morning we're satisfied he's fit to return.
"He's obviously raring to go and from what I've seen on video and the feedback we’ve received, he's certainly not been holding himself back in training. He was really getting stuck in yesterday.
"It’s a massive boost for the Lions to welcome a player of Alun Wyn's stature back."
Following high-scoring wins in their first three matches against franchise sides, the Lions face what is expected to be the toughest examination of their tour so far when they take on South Africa A in Cape Town on Wednesday.