The Championship side emerged as late contenders for McBurnie, who has been used 11 times in the Premier League this season and once in the EFL Cup.
Despite McBurnie playing a respectable 269 minutes of top-flight football this term, Swansea were open to loan offers for the 21-year-old and are said to have attracted interest from a few sides.
Rangers were viewed as the favourites to win the race, but due to confusion over Alfredo Morelos they could not commit, allowing Championship duo Barnsley to steal a march.
According to Sky Sports News, the Tykes have now reached an agreement with the Swans to take boyhood Gers fan McBurnie for the remainder of the season.
Barnsley are expected to cover the majority of the Scotland Under-21s international's £20,000-a-week wages.