The 31-year-old Australian was hurt during the Chiefs' narrow Champions Cup loss at Ulster and needs surgery.
Salvi joined the Chiefs in the summer of 2015 from Leicester and became a key part of the side that made last season's Premiership final.
"We're still waiting to learn the full extent of it but it looks relatively serious," head coach Rob Baxter said.
Salvi was twice short-listed for the Premiership's Player of the Year award while at Leicester and is a former Australia Under-21 captain.
"He does put in a huge contribution over 80 minutes, and that's probably what we'll miss the most," Baxter told BBC Sport.
"His biggest quality is his consistency. He turns up every week, and even when we've had some poor performances and even when some individuals within the side have played poorly, Julian's contribution has always been absolutely first rate," added Baxter.
"He's virtually always our top tackler, virtually always our top clearout guy, and probably the thing that people don't appreciate about him is how important he is to our attack.
"A lot of people talk about open-sides as being tacklers and defenders and disruptive people, but Julian is very important to our attack because he clears so many breakdowns."
Exeter flanker Julian Salvi has been ruled out for the rest of 2016 after injuring his biceps.