Swans boss Michael Laudrup looks set to turn his attention to Odemwingie as he targets a new striker over the summer.
Laudrup is desperate to add more firepower to his ranks ahead of next season's Europa League campaign but has already missed out on a host of targets including Romelu Lukaku, Iago Aspas and Arouna Kone.
The Dane will now try and do a deal for Odemwingie, who is heading for the exit door at the Hawthorns and could be available for as little as £3million.
The Nigerian international is moving into the final year of his contract and West Brom are keen to cash in on a player who publicly attempted to engineer a January move to QPR, while he has also had several spats with the club on Twitter since after being consigned to a bench role under boss Steve Clarke in the second half of last season.
Crystal Palace and Hull City are also showing an interest in the player who hit an impressive 15 Premier League goals in his debut campaign in England back in 2010-11.