Franco Smith revealed the news after the Warriors' Champions Cup victory over Toulon on Friday night.
Speaking to the Offside Line, Smith said: "Jack broke his leg yesterday playing for our A team [in a training match] against Scotland U20s," Smith told .
"He'll be operated on next week. He could recover and still come back for the final two months of the season. That shows how long there is still to go."
Mann has made a single appearance for the Warriors, starring in a win over Bath in last year's Challenge Cup.
The 24-year-old former U20s cap has had a run of bad luck with injury, only recently returning from a lengthy spell on the sidelines through concussion.