Odemwingie has impressed for the Baggies in recent weeks but, along with Youssof Mulumbu, he will take part in the tournament in South Africa which begins on January 19.
"We have good players who can play for a whole month at Albion and they will do well without me. Youssouf is also a very important player for us in the middle but we have good numbers of players," he told Sky Sports News.
"We have [Chris] Brunt, [James] Morrison, [George] Thorne. We have players who can play at this level and Albion took into consideration when recruiting players that a few of us might go.
"I know it's going to be a bit of a loss, us leaving for a few weeks, but we hope the players there can follow the team's performance."
Odemwingie has scored nine goals in 55 games for Nigeria.