The 2009 tourist was injured in a Six Nations match on February 13 and was initially ruled out for up to 12 weeks.
Jones chipped a bone in his left elbow during the first half of Wales' thrilling 31-24 win over Scotland but played on for the remainder of the 80 minutes.
He has since undergone surgery and, although there is no chance of him playing any part in the remainder of the Six Nations, the Ospreys are hopeful he could return in time for the resumption of their European campaign.
The Liberty Stadium outfit travel to San Sebastian to face Biarritz in the last eight on April 10 as they look to become the first Welsh side to be crowned the continent's top dogs.
"He is pushing hard to be available," said Ospreys physio Chris Towers.
"But this is very much a 'best-case' scenario and we will assess his chances again nearer the time.
"[He] has undergone surgery to repair the ligament damage and remove some debris from the joint, and has already started his intensive rehab program following the op."
Jones played in all three Tests on the Lions' summer tour of South Africa, as well as the provincial games against the Royal XV, the Golden Lions and the Sharks.