The two clubs are understood to have agreed an £8m fee for the player last week, yet, per Sky Sports, their failure to convince Clucas to accept personal terms has seen the deal fall through.
The Clarets have been trying to get the 27-year-old on their books for about a year now and made two unsuccessful bids for his services last summer while he was still at Hull City - the midfielder instead opted to move to the Liberty Stadium.
Things haven't worked out as well as he may have imagined though, with the Welsh side having dropped to the Championship, England's second tier, at the end of last season. Clucas also missed out on a chunk of the Swans' campaign with a knee injury, which he is still recovering from.
It was expected that the player would be playing his football at Turf Moor next season and he reportedly underwent a medical with the Clarets last week. However, the move is now off after a breakdown in negotiations.
Clucas, who hails from Lincolnshire, began his youth career with Leicester City aged just 10. He spent eight years with the Foxes before moving to Nettleham and then Lincoln City.
There were moves to Jerez Industrial, Hereford United, Mansfield Town and Chesterfield before he got his big break with Hull in 2015, helping them gain promotion to the top flight via the Championship play-offs a year later.
A return to the Premier League seemed close, but unless Burnley return with better terms or another club come knocking, Clucas could remain in the Championship this year.