Rumours continue to link the injury-prone Bobby Zamora with a move away from Craven Cottage, with fellow forward Andy Johnson also reported to be heading for the exit.
Odemwingie surprised everyone last year after scoring 15 Premier League goals in his debut season at the Hawthorns.
However, even after his overwhelming 2011 success, the forward’s relationship with Hodgson is understood to have become more and more strenuous over recent months and he could be interested in a move away.
The Baggies striker hasn’t replicated the form of last season this campaign, but isn’t dropping off the Premier League standard quite yet.
Despite struggling with a few minor injuries, Odemwingie has started twelve of his sixteen Premier League appearances this campaign, adding four goals to his West Brom tally.
The frontman is also a veteran on the international scene, making over 50 appearances for the Nigerian national side, including a run of displays in the 2010 World Cup.
The Cottagers aren’t the only team interested in Russian-born Odemwingie, and is reports are to be believed, Aston Villa and Everton are also looking at the forward in an attempt to buff up their attacking options.
The deal is expected to be worth £5m, which Fulham will be hoping to raise by selling some fringe players this month.