The 30-year-old felt pain in his right knee during training on Monday and was sent for further tests.
Heynckes says the Chilean now requires a small operation to remove "a loose body" from within the joint, meaning he faces a spell on the sidelines.
"Arturo had a trip without any contact from anyone else," Heynckes told a news conference. "He has a loose body in the knee joint. He needs a small procedure and will be out for the time being.
"He's a fighter. I'm confident for his recovery."
The news is a blow for Bayern, with Vidal set to miss the DFB-Pokal semi-final against Bayer Leverkusen and almost certainly the Bundesliga trip to Hannover at the weekend.
He is also a major doubt for the first leg of their clash with Madrid, which takes place a week on Thursday (AEST).
Bayern could also be without Corentin Tolisso for the trip to the BayArena on Tuesday, with the France international having picked up a problem in the 5-1 thrashing of Borussia Monchengladbach on Saturday.
"Coco picked up another knock against Gladbach and has some bruising. It's not yet clear if he'll make tomorrow's squad," added Heynckes.