Odemwingie was back to his best during the second half against Aston Villa and scored his fifth goal of the campaign to secure a 2-2 draw.
Clarke said: "No-one is going out in January. We don't need to sell anyone.
"Will he move if there is a silly offer? You don't know our chairman (Jeremy Peace). No-one is going out.
"The quicker we get to February 1 the better. No-one is coming in and no-one is going out."
Clarke praised the second half comeback by his side after an indifferent opening 45 minutes.
Goals from Christian Benteke - his 10th of the season - and Gabriel Agbonlahor gave Villa a clear advantage at the interval.
But they were left to regret spurning other chances as Albion hit back through Chris Brunt and Odemwingie.
Clarke said: "I'm not sure relief was the emotion. The emotion was disappointing at the way we played in the first half.
"We knew Villa are a dangerous team but we didn't show our qualities in the first half. We didn't pass the ball like we can and it was nice to get in at half-time and regroup. The second half was much improved and there was a little bit of pride at the way we played."