A price in the region of £2.5 million has been mentioned by sources close to the Russian club. He is expected to sign a three-year deal
Albion are now in the final throes of sorting out his work permit, which should be through later this week.
Odemwingie claims he turned down Greek club Panathinaikos to join Roberto Di Matteo's men.
"I'll be a West Brom player soon," Odemwingie confirmed. "I opted for West Bromwich because for a long time I have wanted to play in England.
"But I thank Panathinaikos for their interest. They are a big club but I decided on West Bromwich. I returned from England on Sunday with a contract which I'd already signed. "It just remains for the clubs to come to an agreement, and my dream to play in the English Premier League will come to fruition.
"The work permit application should be completed later this week and I expect to sign a contract for three years at least. I am back in Moscow now but will return to England this week to finalise the arrangements and begin my new life in England.
"West Bromwich Albion are a good club and I am looking forward to playing for them in the Premier League. This is a new and exciting challenge for me."