The Welsh international dislocated his right shoulder in the second Test defeat to South Africa on June 27 and has since undergone surgery to correct the problem.
The original prognosis suggested a six-month recovery period for the 28-year-old but Ospreys physio Chris Towers has explained that his rehabilitation is progressing as well as can be expected.
"Adam is now five weeks post-op and there have been no complications and none are envisaged as he progresses his rehab," said Towers.
"He will see a specialist next week, and we are anticipating that Adam will then be able to come out of a sling and focus on getting some movement back as well as general strengthening of the joint.
"Generally speaking, you are looking at a six-month rehab process for this kind of injury and operation. If all goes well, we are hopeful of improving on that but it is far too early to make any kind of predictions and we will reassess the situation as we go along."
Jones, who was back at the Ospreys yesterday but unable to take any part in training, is confident that he will be playing again sooner rather than later as he looks to build on a reputation that was greatly enhanced by his performances for Britain and Ireland's elite.
"It's gone well so far, and looking forward, I'm in the hands of the Ospreys medical team for the next four or five months so I'm looking forward to spending lots of hours with Chris Towers, working closely with him and getting to know him very well!" explained Jones.
"I couldn't be in better hands. I just want to get back playing rugby and I'm confident that the team here will have me back on the pitch as good as new soon."