The midfielder was a regular for the Swans after he joined from Hull City in the summer, making 29 appearances in the league, but failed to prevent the club being relegated from the Premier League.
It seemed for a long while that he would make a return to the top flight straight away, after Sean Dyche's Burnley expressed interest in bringing him in to the club.
While a fee of £8 million was agreed between Swansea and the 'Clarets', 'Sky Sports' report that the deal will now not go ahead after the player and club failed to agree on the personal terms in the contract.
This means that Burnley may need to quickly find an alternative to Clucas, with the transfer window due to shut on 9th August and the club having failed to significantly improve their squad.
Defender Ben Gibson has arrived at Turf Moor, but they have failed to add any other signings so far.
European football will add an extra burden to the squad, with the extra games potentially affecting the club's league form should they fail to add to their squad depth.