The 22-year-old forward has been at Oakwell for six years and played 82 games, scoring 12 goals.
He was a regular starter up front last season and scored three in 39 starts but his nine assists were the sixth most in the Championship.
A Barnsley statement said: "The decision to sell Brown has been made reluctantly, with the Yorkshire born forward putting pen to paper on a three-year-deal in 2019.
"Monies received upfront will be immediately reinvested into the team, as Gerhard and his squad set out on another Sky Bet Championship campaign. Barnsley Football Club would like to deeply thank Jacob for his efforts whilst at Oakwell and wish him well in his future career."