Clarke has told the Nigerian international to "say it to my face" if he has got any grievances against him or the club.
Odemwingie complained on Twitter about being out of the starting line-up after the goalless draw at Stoke, hinted team-mates were talking about him behind his back and conceded "his wish" was to be sacked.
It followed on from his previous social networking attacks during January when his transfer request was rejected and he failed to secure a deadline day move to QPR.
Clarke admits he is fed up of talking about the 31-year-old when facing the media today ahead of Saturday's visit to West Ham.
He said: "I haven't made very much of it. I don't read Twitter. I understand there were some more comments on Twitter.
"I told Richard Garlick (sporting and technical director) and the club to deal with it in the manner they see fit.
"I think if he (Odemwingie) has got something to say to me, he should come and say it to my face.
"I haven't got anything to say to him. If he's got anything to say to me, he should come and say it to my face. He is the one who has the grievance, not me.
"But I'm not interested in what he said. The club will deal with that side of it.
"Are the club taking extra action? I don't know because Richard has been off ill. He hasn't been in."
When asked about the behaviour of one of his leading players, Clarke said: "Sorry, he's what? He is one of my players. That's it, he is one of my players.
Albion are lying eighth in the Barclays Premier League and still have an outside chance of qualifying for Europe.
But Clarke said: "I'm just disappointed I have to spend so much time talking about one player (Odemwingie) when I've got a squad of players who have made a fantastic contribution this season.
"I'm fed up talking about Odemwingie. You've asked about it, I've answered it, let's move on. I'm not answering any more questions on Peter."
Clarke confirmed Odemwingie was in the squad to face West Ham on Saturday.