West Ham will reportedly listen to offers for Antonio, who has been linked with a move to Crystal Palace.
And it was Palace who led the betting for his signature until Friday evening, when Burnley's odds suddenly dropped from 14/1 to as short as 3/1.
Burnley are already a winger light going into the summer, after George-Kevin N'Koudou's return to Tottenham Hotspur.
And it is possible the Clarets would be interested in a move for Antonio, who has scored or made 32 goals in 90 games at Premier League level and been called up by England on more than one occasion without being capped.
Crystal Palace are 7/2 to sign Antonio, ahead of Chelsea (16/1), Wolves (20/1) and Cardiff (25/1).
You can get odds of 5/6 on the 28-year-old either leaving or remaining at West Ham.
Could Burnley move for the West Ham winger?