The 31-year-old striker is wanted by Queens Park Rangers and added to his reputation by scoring the equaliser in West Brom's 2-2 draw with Aston Villa on Sunday.
Manager Steve Clarke has already signalled his determination to keep hold of the Nigerian, and goalkeeper Foster has echoed those sentiments.
He said: "We can't afford to let anybody go. We have been struggling [with injuries] - we have had 13 or 14 players in training at times. Hopefully he [Odemwingie] won't be going anywhere."
Odemwingie joined West Brom in a £2m deal from Lokomotiv Moscow in 2010 and has gone on to score 31 goals in 83 appearances.
Although he can often be as frustrating as he is valuable, Foster feels losing his natural talent would be a tough gap to fill.
He added: "He has always got that ability to score goals for us when it matters. He can be frustrating at times, but other times he can be magical.
"Sometimes everyone says 'just try to shoot'. I know sometimes it will go wide or over the bar, but he can unleash and when he does, they can be unstoppable.
"That's the beauty of him, he can frustrate you, but has so much skill and ability."