The Russian forward has not started a game for the Baggies since he drove to London on transfer deadline day in an attempt to force a move to Queens Park Rangers.
However, Clarke claims that Odemwingie could still see out the remaining 16 months of his current contract at The Hawthorns.
"He could have a future beyond the summer," Clarke told the Birmingham Mail. "He has a year left on his contract.
"Things happen and life moves on – that's the world. It happens in life and in sport.
"There is a possibility [of him staying], but I haven't spoken to him. He's still trying to get back into the team. He is under contract to the club for another year."
Odemwingie has scored 30 goals in 82 Premier League appearances for West Brom.