The former Serbia skipper, 34, had surgery at the end of last term and was aiming to recover in time for the new Serie A season.
He returned to pre-season training in July after Inter claimed the operation had been a success, but has now suffered a setback in his recovery.
"It is a problem that has long plagued me and was not completely solved with the first surgical procedure," Stankovic, who was linked with a summer move to Premier League club West Ham, told SFF.
"In consultation with the medical team at Inter, as well as on the basis of new detailed checks, we decided to go for further surgery. This was the only good solution.
"Previously, I did not want to talk in public because I was waiting for only one thing - to get back on the pitch. Unfortunately, I will have to wait a little longer."