Rays manager Kevin Cash said the left-hander will head to the injured list and have an MRI on Wednesday.
It's a potentially significant injury for the Rays, who lost their fourth straight game since clinching a playoff berth.
Poche is 4-2 with a 3.99 ERA in 65 relief appearances spanning 58⅔ innings. He entered the game with a 26.3% strikeout rate, 8.8% walk rate and 32.5% ground ball rate.
"He's crushed. We're all crushed for him,'' Cash told reporters in Boston. "He's such a big part of our bullpen, and for it to happen the way it happened -- it's unfortunate.''
The Rays will be the American League's No. 6 seed and get a wild-card matchup with the AL Central champion Cleveland Guardians.
Information from The Associated Press was used in this report.