The Whites are keen to land the right-back on a permanent transfer after he impressed during 14 Championship starts at the back end of last season.
Richards, who made his Swansea debut in 2009, has a year left on his current contract at the Liberty Stadium and wishes to leave to play regular first-team games.
It is understood other clubs are interested by Fulham are in a commanding position and negotiations continue.
Richards has played in the Premier League for the Swans and has made 42 appearances for the club altogether, whilst also previously spending time on loan at Crystal Palace and Huddersfield.
Sky sources also understand the Whites are keen to land free agent Ben Pringle, who left Rotherham at the end of the campaign, on a free transfer.
Pringle, 25, has been an instrumental part of Rotherham’s success in recent years and scored three times in 43 appearances in their inaugural Championship season.