Yang, also vault world champion in 2011 and 2013, suffered the injury during floor practice at the Taeneung National Training Center in Seoul on Tuesday and had surgery the following day.
"The rehab is expected to take three to four months," Han Chung-sik, a Korea Gymnastic Association official, told the Yonhap news agency.
"Even if Yang recovers before the Olympics, he will not have had enough training to compete.
"He may demonstrate skills in the air, but it will be difficult for him to perform at the Olympics if he can't land well."
The 23-year-old Yang, who had been practising for the first round of the national trials next month, soared to the top of the world in the vault on the back of his signature triple-twisting front somersault, which is named after him.
It was the successful execution of this manoeuvre, which carried the highest level of difficulty coefficient, that set him on his way to the gold medal as a teenager at the London Olympics.
Yang, then a student who supported his impoverished parents from his Korea Gymnastic Association stipend, has been regularly sidelined by injuries since his London triumph, however.
A second, unnamed, Korean gymnastics official told Yonhap that Yang had sustained the injury doing routine floor work, not attempting a complicated manoeuvre.
"Yang was managing his condition more carefully than he did for the 2012 London Games," the official added.
"He is hurt and feels sadder than anyone else."
North Korean Ri Se-gwang, who took Yang's vault title at the 2014 world championships and retained the crown in Glasgow last year, will be favourite to strike Olympic gold in Rio in August.
(Reporting by Ian Ransom in Melbourne and Nick Mulvenney in Sydney; Editing by Greg Stutchbury)
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