Thornton, 38, came off the ice with 40 seconds left in the third period. He was hurt as he was tumbling to the ice and teammate Mikkel Boedker fell on his right leg.
Thornton had major surgery to his left knee after last season, during which he tore his MCL and ACL.
Sharks coach Peter DeBoer said he expected an update on Thornton's condition Wednesday morning but said he didn't like the looks of it.
"He doesn't leave [the ice] unless, you know -- we all know him," DeBoer said. "Not a good feeling."
Thornton has 13 goals and 23 assists in 47 games this season, his 20th in the NHL, and has been hot lately, with nine points in his past seven games. He's second among active players in career points with 1,427, trailing only Jaromir Jagr of the Calgary Flames (1,921).
The Sharks lost to the Jets 5-4 in overtime and are second in the Pacific Division with a 26-14-7 record.
Sharks captain Joe Pavelski said his team can withstand an injury to Thornton.
"We have a much better chance with him in the lineup," Pavelski said, "but if he's going to miss a little bit of time, someone's going to have to step up and get the job done."