Saros looked to be limping as he skated off of the ice and headed to the dressing room with 6:32 remaining in the third period.
"I don't have any update on him," Hynes told reporters after the game. "He's still getting evaluated. I haven't talked to the trainer or anything yet, so we'll probably have more on that tomorrow."
David Rittich came on in relief and gave up the game-tying goal from Matthew Tkachuk with a second left on the clock in regulation and conceded the winner in overtime as Elias Lindholm found the back of the net as Calgary completed a 5-4 comeback victory.
Saros, who played in his first All-Star Game this season, has posted a career-high 38 wins during the 2021-22 campaign as well as a 2.64 goals-against average and .918 save percentage.
Nashville (44-29-7) has clinched a playoff berth and has two regular-season games remaining.
The Predators are tied with the Dallas Stars in the Western Conference wild-card standings with 95 points each.