The Nigerian striker is actively seeking a move away from the Hawthorns after handing in a transfer request on Friday which was rejected by the club, with Harry Redknapp interested
The Baggies' sporting and technical director Richard Garlick told the club's official website: "QPR have made a second, improved offer for Peter, which we have turned down.
"As we have repeatedly stated, we have no need or desire to sell our core players.
"Since we turned down Peter's transfer request, he has made his position clear in a statement to the media and via his personal Twitter account.
| QPR LATEST
|18/1||QPR are 18/1 to beat Manchester City 1-0 with bet365
The 31-year-old striker's actions on the social media site could see him fined two weeks' wages, approximately £70,000 after he tweeted: "Best compliment I have ever received in my career as a player. Manager like HR [Harry Redknapp] spending to bring me for mission 'impossible'."
QPR boss Redknapp is eager to add the Nigerian international to his squad before Thursday's deadline in the wake of the Rs' humiliating FA Cup exit to MK Dons on Saturday.
Odemwingie has made 80 appearances for the Baggies, scoring 30 goals since his move Lokomotiv Moscow in 2010.