The 35-year-old, who was set to captain Warren Gatland's touring party in South Africa, suffered the injury against Japan at the end of June but could still fly out on Tuesday, subject to medical confirmation.
The Lions face South Africa A, a side packed with 11 World Cup winners, on Wednesday and Gatland revealed in Monday's news conference that Jones could well return.
"A few weeks ago we didn't think there was an option," the Lions head coach said. "We're waiting for medical reports from him to see what happens with that.
"He's been back fully training with Wales – he's training this week with them. I think he's doing a double session with them tomorrow and we'll see what the outcome of that is."
Conor Murray, who is yet to start a game on tour for the Lions, came in to act as Jones' replacement in the captaincy role.
However, Gatland appreciates the invaluable role the Welshman could play if fit.
"He has made a remarkable recovery in terms of that injury, which wasn't as bad as they first thought," he added.
"It would be a boost to the squad having someone of his experience and calibre to come back into the squad."
Experience is exactly what Jones brings, as he represents the only member of the current touring squad to have faced the Springboks with the Lions previously, while also being the most-capped international of all time.
Having triumphed comfortably in the three pre-test matches against South African club sides the Lions and the Sharks, Gatland's team face South Africa A in what is being billed as an unofficial fourth test.
A subsequent fixture against DHL Stormers will follow before the three showpiece clashes with the Springboks get underway on the 24th July, by which time Gatland will be hoping Jones is fit.