The 29-year old Nigerian has attracted plenty of interest following his 15-goal debut Premier League season with the Gunners putting down an offer this week.
"Luckily they are under contract, so that's not a great problem for us," said Hodgson.
"We are working very hard to make sure they are happy at the club, and we will do our very best to ensure that. We are a club with a very healthy economy. The chairman has a very clear idea of the direction that the club wants to go in. He is certainly more interested in retaining our players than selling them.
"So players like Odemwingie and Brunt, who have long-term contracts, I expect them to stay at the club."