Swansea expressed an interest in Sylvestre last summer, but Blackpool rebuffed their efforts to sign the playmaker.
Sylvestre is out of contract this summer at Bloomfield Road and he has failed to agree a new deal at the club.
The news has alerted a host of clubs to Sylvestre's availability on a free transfer, with the likes of Fulham and Crystal Palace credited with an interest in the 29-year-old in the past.
Swansea boss Michael Laudrup is thought to be a big admirer of Sylvestre and is keen to bring him to the Liberty Stadium to bolster his squad as the club prepare for the extra demands of playing in Europe next term.
Sylvestre has been at Bloomfield Road since 2010, having played for Barcelona earlier in his career and also spent time at Sparta Prague and Mlada Boleslav.