Hillingdon-born Mawson joined Barnsley from Brentford in June 2015 after the player had loan spells at Luton, Welling and Wycombe.
Mawson has made 64 appearances and last season was a part of the Barnsley side which won the Sky Bet League One play-offs and lifted the Football League trophy at Wembley.
"The move is subject to the 22-year-old agreeing personal terms at the Liberty," said a statement on the official Swansea website.
Swansea manager Francesco Guidolin had targeted a central defender following the departure of club captain Ashley Williams to Everton.
Jordi Amat, Federico Fernandez and former Ajax youngster Mike van der Hoorn are currently the only three central defenders at Swansea.
"We need another centre-back," Guidolin had said ahead of Saturday's Premier League game at Leicester.
"I am happy with the three centre-backs I have, but the season is very long so we need another one."
Source : PA
Source: PA