The Evening Post says that Brendan Rodgers is weighing up a move for the former Manchester United hotshot, who is out of contract next summer.
It's thought that Eagles is free to leave Turf Moor for the right price, with the club looking to recoup at least the £1m paid to United three years ago.
Swansea have been linked with a host of widemen this summer, with Wayne Routledge and Marvin Emnes thought to be highest on the list of targets.
But with Middlesbrough demanding £4m for both Emnes and Leroy Lita, along with the competition for Routledge's signature, a move for Eagles could be a better option for Rodgers.
The winger scored 11 times last season as the Clarets missed out on a play-off spot.