His potential arrival could hasten the departure of Bafetimbi Gomis.
The Swans are looking to add a striker to their squad in the January transfer window, having struggled in that area this season.
They have managed just 22 goals in 23 games so far this season, the joint second worst record in the division.
Andre Ayew has managed eight goals for the side this season, with Bafetimbi Gomis their highest scoring out and out striker with five to his name.
They have been linked with several stars already this month, but are now thought to be leading the race for Gayle, according to Sky Sports.
The Eagles are unlikely to let their striker leave on the cheap though, having paid Peterborough £6milllion for his services back in 2013.
Should Gayle arrive at the Liberty Stadium, it is likely to hasten the departure of Gomis from the club.
The striker has been linked with an exit from the club this month, with Newcastle United, who are also keen to add a striker to their squad, thought to be considering a £7million for him, as reported by the Daily Mail.