The 23-year-old winger has struggled to play regular football since arriving at Oakwell in January 2017, with just 13 of his 52 league appearances being starts.
The Wales international started only five League One matches in the 2018/19 season, as the Reds gained promotion, but was still offered a contract extension. In total, he played 59 games for the club and scored four goals.
He is the second player from last season's Barnsley squad to move north of the border in the last week after centre-back Adam Jackson joined Paul Heckingbottom's Hibernian following his release by the Reds.
Hedges' fellow Wales player Adam Davies is also likely to turn down a new deal, with a move to Stoke City expected to be announced soon.