The Swans are keen to bring the Ghana international, 26, whose contract at Oakwell expires at the end of the season, to the Liberty Stadium.
He has rejected a new deal and Barnsley are set to cash in on him in January.
Barnsley had originally been asking for £3million for the attacking full-back but they are now willing to drop their price and MirrorSport understands Swansea have started to negotiate.
Long-term admirers Swansea are moving quickly to try to head off rival top-flight interest from West Brom, who have been scouting the defender.
Yiadom nearly moved to promoted Huddersfield in the summer but a deal was aborted in the last hour due to injury concerns.
He has been capped twice by Ghana. Read Full Story
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