The Nigerian international has been in great form since arriving from Lokomotiv Moscow and has understandably attracted interest from bigger clubs.
Hodgson is keen to retain his services but is well aware that at a club the size of West Brom, every player has a price and if a big offer came in then he would be forced to consider it.
He told the Birmingham Mail: "He's a good player and we would like to keep him here.
"Whether we receive any offers or not, who can say? But I'm not sure there's an awful lot of money around.
"Certainly we wouldn't be looking to get rid of him so it would have to be the type of offer you can't refuse and we know what top front players cost in the market.
"I'm not suggesting £50 million but they certainly cost top money, so it would be a question of what teams are out there with that sort of money to spend and whether Peter Odemwingie is the type of player they are looking for, so there are a lots of ifs and buts."
Italian giants Juventus have been linked with a move for Odemwingie, but as a man who has worked in both England and Italy, Hodgson is confident these rumours are unfounded.
He continued: "I don't take a lot of notice of the Juventus thing. Any agent in the world can always drum up an Italian club.
"When I was at Inter there wasn't a day that went by when there weren't 10 players we were supposedly signing.
"It would have run into thousands over two years.
"When I hear about Juventus I just shrug my shoulders. If you're an English journalist thinking 'who can I link him with?' Inter and Juventus are always good bets."
With the Baggies looking safe in the Premier League, the former Fulham boss is starting to plan for next season and sees Odemwingie as a big part of his team going forward.
"Peter seems perfectly happy here and he's a joy to work with and I think he has enjoyed his time here," he said.
"I don't think he's looking to take the first opportunity to get out of the club.
"We're not going to have a chance to do well next year if we've got to start building again.
"Our only chance to do better next year is to build on what we've got.
"If we're going to lose the best ones we've got and start again then I would think the task is going to be more than uphill.
"No-one wants to be in that situation."