Burns has been a regular in the Tykes side this season but was withdrawn from the squad for Tuesday's game at Bournemouth and is expected to sign a two-and-a-half-year deal at Wisla after the clubs agreed a fee.
The Australia international has hopes of playing in this summer's World Cup in Germany and will play under former Chelsea and Sheffield Wednesday man Dan Petrescu at Krakow, who have won the Polish Championship for the last three years.